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Metamath Proof Explorer

Theorem rloccring

Description: The ring localization L of a commutative ring R by a multiplicatively closed set S is itself a commutative ring. (Contributed by Thierry Arnoux, 4-May-2025)

Ref Expression
Hypotheses rlocaddval.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
rlocaddval.2 ⊢ · = ( .r ‘ 𝑅 )
rlocaddval.3 ⊢ + = ( +g ‘ 𝑅 )
rlocaddval.4 ⊢ 𝐿 = ( 𝑅 RLocal 𝑆 )
rlocaddval.5 ⊢ ∼ = ( 𝑅 ~RL 𝑆 )
rlocaddval.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
rlocaddval.s ⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
Assertion rloccring ( 𝜑 → 𝐿 ∈ CRing )

Proof

Step Hyp Ref Expression
1 rlocaddval.1 ⊢ 𝐵 = ( Base ‘ 𝑅 )
2 rlocaddval.2 ⊢ · = ( .r ‘ 𝑅 )
3 rlocaddval.3 ⊢ + = ( +g ‘ 𝑅 )
4 rlocaddval.4 ⊢ 𝐿 = ( 𝑅 RLocal 𝑆 )
5 rlocaddval.5 ⊢ ∼ = ( 𝑅 ~RL 𝑆 )
6 rlocaddval.r ⊢ ( 𝜑 → 𝑅 ∈ CRing )
7 rlocaddval.s ⊢ ( 𝜑 → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
8 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
9 eqid ⊢ ( -g ‘ 𝑅 ) = ( -g ‘ 𝑅 )
10 eqid ⊢ ( 𝐵 × 𝑆 ) = ( 𝐵 × 𝑆 )
11 eqid ⊢ ( mulGrp ‘ 𝑅 ) = ( mulGrp ‘ 𝑅 )
12 11 1 mgpbas ⊢ 𝐵 = ( Base ‘ ( mulGrp ‘ 𝑅 ) )
13 12 submss ⊢ ( 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) → 𝑆 ⊆ 𝐵 )
14 7 13 syl ⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
15 1 8 2 9 10 4 5 6 14 rlocbas ⊢ ( 𝜑 → ( ( 𝐵 × 𝑆 ) / ∼ ) = ( Base ‘ 𝐿 ) )
16 eqidd ⊢ ( 𝜑 → ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 ) )
17 eqidd ⊢ ( 𝜑 → ( .r ‘ 𝐿 ) = ( .r ‘ 𝐿 ) )
18 eqid ⊢ ( Base ‘ 𝐿 ) = ( Base ‘ 𝐿 )
19 eqid ⊢ ( 0g ‘ 𝐿 ) = ( 0g ‘ 𝐿 )
20 eqid ⊢ ( +g ‘ 𝐿 ) = ( +g ‘ 𝐿 )
21 6 crngringd ⊢ ( 𝜑 → 𝑅 ∈ Ring )
22 1 8 ring0cl ⊢ ( 𝑅 ∈ Ring → ( 0g ‘ 𝑅 ) ∈ 𝐵 )
23 21 22 syl ⊢ ( 𝜑 → ( 0g ‘ 𝑅 ) ∈ 𝐵 )
24 eqid ⊢ ( 1r ‘ 𝑅 ) = ( 1r ‘ 𝑅 )
25 11 24 ringidval ⊢ ( 1r ‘ 𝑅 ) = ( 0g ‘ ( mulGrp ‘ 𝑅 ) )
26 25 subm0cl ⊢ ( 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) → ( 1r ‘ 𝑅 ) ∈ 𝑆 )
27 7 26 syl ⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ 𝑆 )
28 23 27 opelxpd ⊢ ( 𝜑 → ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
29 5 ovexi ⊢ ∼ ∈ V
30 29 ecelqsi ⊢ ( ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
31 28 30 syl ⊢ ( 𝜑 → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
32 31 15 eleqtrd ⊢ ( 𝜑 → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( Base ‘ 𝐿 ) )
33 15 eleq2d ⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ↔ 𝑥 ∈ ( Base ‘ 𝐿 ) ) )
34 33 biimpar ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐿 ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
35 simpr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
36 35 oveq2d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
37 21 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑅 ∈ Ring )
38 7 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
39 38 13 syl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑆 ⊆ 𝐵 )
40 simplr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑏 ∈ 𝑆 )
41 39 40 sseldd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑏 ∈ 𝐵 )
42 1 2 8 37 41 ringlzd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) · 𝑏 ) = ( 0g ‘ 𝑅 ) )
43 simpllr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑎 ∈ 𝐵 )
44 1 2 24 37 43 ringridmd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑎 · ( 1r ‘ 𝑅 ) ) = 𝑎 )
45 42 44 oveq12d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) = ( ( 0g ‘ 𝑅 ) ( +g ‘ 𝑅 ) 𝑎 ) )
46 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
47 37 ringgrpd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑅 ∈ Grp )
48 1 46 8 47 43 grplidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) ( +g ‘ 𝑅 ) 𝑎 ) = 𝑎 )
49 45 48 eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) = 𝑎 )
50 1 2 24 37 41 ringlidmd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 1r ‘ 𝑅 ) · 𝑏 ) = 𝑏 )
51 49 50 opeq12d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ = ⟨ 𝑎 , 𝑏 ⟩ )
52 51 eceq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
53 6 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑅 ∈ CRing )
54 23 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 0g ‘ 𝑅 ) ∈ 𝐵 )
55 38 26 syl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 1r ‘ 𝑅 ) ∈ 𝑆 )
56 1 2 46 4 5 53 38 54 43 55 40 20 rlocaddval ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( ( ( 0g ‘ 𝑅 ) · 𝑏 ) ( +g ‘ 𝑅 ) ( 𝑎 · ( 1r ‘ 𝑅 ) ) ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ )
57 52 56 35 3eqtr4d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = 𝑥 )
58 36 57 eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = 𝑥 )
59 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
60 59 elrlocbasi ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ∃ 𝑎 ∈ 𝐵 ∃ 𝑏 ∈ 𝑆 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
61 58 60 r19.29vva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = 𝑥 )
62 34 61 syldan ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐿 ) ) → ( [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) 𝑥 ) = 𝑥 )
63 1 2 3 4 5 53 38 43 54 40 55 20 rlocaddval ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
64 35 oveq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) )
65 44 42 oveq12d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) = ( 𝑎 + ( 0g ‘ 𝑅 ) ) )
66 1 3 8 47 43 grpridd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑎 + ( 0g ‘ 𝑅 ) ) = 𝑎 )
67 65 66 eqtr2d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑎 = ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) )
68 1 2 24 37 41 ringridmd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑏 · ( 1r ‘ 𝑅 ) ) = 𝑏 )
69 68 eqcomd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑏 = ( 𝑏 · ( 1r ‘ 𝑅 ) ) )
70 67 69 opeq12d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ )
71 70 eceq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ = [ ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
72 35 71 eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑥 = [ ⟨ ( ( 𝑎 · ( 1r ‘ 𝑅 ) ) + ( ( 0g ‘ 𝑅 ) · 𝑏 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
73 63 64 72 3eqtr4d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
74 73 60 r19.29vva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
75 34 74 syldan ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( Base ‘ 𝐿 ) ) → ( 𝑥 ( +g ‘ 𝐿 ) [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
76 18 19 20 32 62 75 ismgmid2 ⊢ ( 𝜑 → [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ = ( 0g ‘ 𝐿 ) )
77 simp-4r ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
78 simpr ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
79 77 78 oveq12d ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
80 6 ad8antr ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑅 ∈ CRing )
81 7 ad8antr ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
82 simp-6r ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑎 ∈ 𝐵 )
83 simpllr ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑐 ∈ 𝐵 )
84 simp-5r ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑏 ∈ 𝑆 )
85 simplr ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑑 ∈ 𝑆 )
86 1 2 3 4 5 80 81 82 83 84 85 20 rlocaddval ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
87 80 crnggrpd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑅 ∈ Grp )
88 21 ad8antr ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑅 ∈ Ring )
89 81 13 syl ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑆 ⊆ 𝐵 )
90 89 85 sseldd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑑 ∈ 𝐵 )
91 1 2 88 82 90 ringcld ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑎 · 𝑑 ) ∈ 𝐵 )
92 89 84 sseldd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑏 ∈ 𝐵 )
93 1 2 88 83 92 ringcld ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑐 · 𝑏 ) ∈ 𝐵 )
94 1 3 87 91 93 grpcld ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) ∈ 𝐵 )
95 11 2 mgpplusg ⊢ · = ( +g ‘ ( mulGrp ‘ 𝑅 ) )
96 95 81 84 85 submcld ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) ∈ 𝑆 )
97 94 96 opelxpd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
98 29 ecelqsi ⊢ ( ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
99 97 98 syl ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
100 86 99 eqeltrd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
101 79 100 eqeltrd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
102 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
103 102 elrlocbasi ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∃ 𝑐 ∈ 𝐵 ∃ 𝑑 ∈ 𝑆 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
104 101 103 r19.29vva ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
105 simplr ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
106 105 elrlocbasi ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ∃ 𝑎 ∈ 𝐵 ∃ 𝑏 ∈ 𝑆 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
107 104 106 r19.29vva ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
108 107 3impa ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
109 6 ad10antr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑅 ∈ CRing )
110 109 crnggrpd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑅 ∈ Grp )
111 21 ad10antr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑅 ∈ Ring )
112 simp-9r ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑎 ∈ 𝐵 )
113 7 ad10antr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑆 ∈ ( SubMnd ‘ ( mulGrp ‘ 𝑅 ) ) )
114 113 13 syl ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑆 ⊆ 𝐵 )
115 simp-5r ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑑 ∈ 𝑆 )
116 114 115 sseldd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑑 ∈ 𝐵 )
117 1 2 111 112 116 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑑 ) ∈ 𝐵 )
118 simplr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑓 ∈ 𝑆 )
119 114 118 sseldd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑓 ∈ 𝐵 )
120 1 2 111 117 119 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) · 𝑓 ) ∈ 𝐵 )
121 simp-6r ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑐 ∈ 𝐵 )
122 simp-8r ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑏 ∈ 𝑆 )
123 114 122 sseldd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑏 ∈ 𝐵 )
124 1 2 111 121 123 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑐 · 𝑏 ) ∈ 𝐵 )
125 1 2 111 124 119 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑏 ) · 𝑓 ) ∈ 𝐵 )
126 simpllr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑒 ∈ 𝐵 )
127 1 2 111 123 116 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) ∈ 𝐵 )
128 1 2 111 126 127 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · ( 𝑏 · 𝑑 ) ) ∈ 𝐵 )
129 1 3 110 120 125 128 grpassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( ( 𝑐 · 𝑏 ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) ) )
130 1 2 111 112 116 119 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) · 𝑓 ) = ( 𝑎 · ( 𝑑 · 𝑓 ) ) )
131 1 2 109 121 123 119 crng32d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑏 ) · 𝑓 ) = ( ( 𝑐 · 𝑓 ) · 𝑏 ) )
132 1 2 109 126 123 116 crng12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · ( 𝑏 · 𝑑 ) ) = ( 𝑏 · ( 𝑒 · 𝑑 ) ) )
133 1 2 111 126 116 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · 𝑑 ) ∈ 𝐵 )
134 1 2 109 123 133 crngcomd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑒 · 𝑑 ) ) = ( ( 𝑒 · 𝑑 ) · 𝑏 ) )
135 132 134 eqtrd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑒 · ( 𝑏 · 𝑑 ) ) = ( ( 𝑒 · 𝑑 ) · 𝑏 ) )
136 131 135 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑐 · 𝑏 ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) )
137 130 136 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( ( 𝑐 · 𝑏 ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) ) )
138 129 137 eqtrd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) ) )
139 1 3 2 111 117 124 119 ringdird ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) = ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) )
140 139 oveq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( ( ( 𝑎 · 𝑑 ) · 𝑓 ) + ( ( 𝑐 · 𝑏 ) · 𝑓 ) ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) )
141 1 2 111 121 119 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑐 · 𝑓 ) ∈ 𝐵 )
142 1 3 2 111 141 133 123 ringdird ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) = ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) )
143 142 oveq2d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) · 𝑏 ) + ( ( 𝑒 · 𝑑 ) · 𝑏 ) ) ) )
144 138 140 143 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) = ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) )
145 1 2 111 123 116 119 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) = ( 𝑏 · ( 𝑑 · 𝑓 ) ) )
146 144 145 opeq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ = ⟨ ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ )
147 146 eceq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ = [ ⟨ ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
148 1 3 110 117 124 grpcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) ∈ 𝐵 )
149 95 113 122 115 submcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) ∈ 𝑆 )
150 1 2 3 4 5 109 113 148 126 149 118 20 rlocaddval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑓 ) + ( 𝑒 · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ )
151 1 3 110 141 133 grpcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ∈ 𝐵 )
152 95 113 115 118 submcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑑 · 𝑓 ) ∈ 𝑆 )
153 1 2 3 4 5 109 113 112 151 122 152 20 rlocaddval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · ( 𝑑 · 𝑓 ) ) + ( ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) · 𝑏 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
154 147 150 153 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
155 1 2 3 4 5 109 113 112 121 122 115 20 rlocaddval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
156 155 oveq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
157 1 2 3 4 5 109 113 121 126 115 118 20 rlocaddval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ )
158 157 oveq2d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
159 154 156 158 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
160 simp-7r ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
161 simp-4r ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
162 160 161 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
163 simpr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ )
164 162 163 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
165 161 163 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
166 160 165 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
167 159 164 166 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
168 simpr3 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
169 168 ad6antr ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
170 169 elrlocbasi ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ∃ 𝑒 ∈ 𝐵 ∃ 𝑓 ∈ 𝑆 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ )
171 167 170 r19.29vva ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
172 simpr2 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
173 172 ad3antrrr ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
174 173 elrlocbasi ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∃ 𝑐 ∈ 𝐵 ∃ 𝑑 ∈ 𝑆 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ )
175 171 174 r19.29vva ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
176 simpr1 ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
177 176 elrlocbasi ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ∃ 𝑎 ∈ 𝐵 ∃ 𝑏 ∈ 𝑆 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
178 175 177 r19.29vva ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( +g ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) )
179 15 16 108 178 31 61 74 ismndd ⊢ ( 𝜑 → 𝐿 ∈ Mnd )
180 eqid ⊢ ( invg ‘ 𝑅 ) = ( invg ‘ 𝑅 )
181 1 180 47 43 grpinvcld ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) ∈ 𝐵 )
182 181 40 opelxpd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ∈ ( 𝐵 × 𝑆 ) )
183 29 ecelqsi ⊢ ( ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
184 182 183 syl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
185 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ )
186 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
187 185 186 oveq12d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
188 187 eqeq1d ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑢 = [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ) → ( ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ↔ ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) )
189 1 2 3 4 5 53 38 181 43 40 40 20 rlocaddval ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ )
190 1 3 8 180 47 43 grplinvd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) + 𝑎 ) = ( 0g ‘ 𝑅 ) )
191 190 oveq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) + 𝑎 ) · 𝑏 ) = ( ( 0g ‘ 𝑅 ) · 𝑏 ) )
192 1 3 2 37 181 43 41 ringdird ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) + 𝑎 ) · 𝑏 ) = ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) )
193 191 192 42 3eqtr3d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) = ( 0g ‘ 𝑅 ) )
194 193 opeq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) , ( 𝑏 · 𝑏 ) ⟩ = ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ )
195 194 eceq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) · 𝑏 ) + ( 𝑎 · 𝑏 ) ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ = [ ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ )
196 1 8 24 2 9 10 5 6 7 erler ⊢ ( 𝜑 → ∼ Er ( 𝐵 × 𝑆 ) )
197 196 ad4antr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∼ Er ( 𝐵 × 𝑆 ) )
198 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ = ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ )
199 eqidd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ = ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ )
200 95 38 40 40 submcld ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑏 · 𝑏 ) ∈ 𝑆 )
201 1 2 24 37 54 ringridmd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
202 39 200 sseldd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑏 · 𝑏 ) ∈ 𝐵 )
203 1 2 8 37 202 ringlzd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) = ( 0g ‘ 𝑅 ) )
204 201 203 oveq12d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) = ( ( 0g ‘ 𝑅 ) ( -g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) )
205 1 8 9 grpsubid ⊢ ( ( 𝑅 ∈ Grp ∧ ( 0g ‘ 𝑅 ) ∈ 𝐵 ) → ( ( 0g ‘ 𝑅 ) ( -g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
206 47 54 205 syl2anc ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 0g ‘ 𝑅 ) ( -g ‘ 𝑅 ) ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
207 204 206 eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) = ( 0g ‘ 𝑅 ) )
208 207 oveq2d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑏 ) · ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) ) = ( ( 𝑏 · 𝑏 ) · ( 0g ‘ 𝑅 ) ) )
209 1 2 8 37 202 ringrzd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑏 ) · ( 0g ‘ 𝑅 ) ) = ( 0g ‘ 𝑅 ) )
210 208 209 eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑏 ) · ( ( ( 0g ‘ 𝑅 ) · ( 1r ‘ 𝑅 ) ) ( -g ‘ 𝑅 ) ( ( 0g ‘ 𝑅 ) · ( 𝑏 · 𝑏 ) ) ) ) = ( 0g ‘ 𝑅 ) )
211 1 5 39 8 2 9 198 199 54 54 200 55 200 210 erlbrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ ∼ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ )
212 197 211 erthi ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( 0g ‘ 𝑅 ) , ( 𝑏 · 𝑏 ) ⟩ ] ∼ = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
213 189 195 212 3eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( ( invg ‘ 𝑅 ) ‘ 𝑎 ) , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
214 184 188 213 rspcedvd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ∃ 𝑢 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
215 214 60 r19.29vva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ∃ 𝑢 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ( 𝑢 ( +g ‘ 𝐿 ) 𝑥 ) = [ ⟨ ( 0g ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ )
216 15 16 76 179 215 isgrpd2e ⊢ ( 𝜑 → 𝐿 ∈ Grp )
217 77 78 oveq12d ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
218 eqid ⊢ ( .r ‘ 𝐿 ) = ( .r ‘ 𝐿 )
219 1 2 3 4 5 80 81 82 83 84 85 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
220 1 2 88 82 83 ringcld ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑎 · 𝑐 ) ∈ 𝐵 )
221 220 96 opelxpd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
222 29 ecelqsi ⊢ ( ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
223 221 222 syl ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
224 219 223 eqeltrd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
225 217 224 eqeltrd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
226 225 103 r19.29vva ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
227 226 106 r19.29vva ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
228 227 3impa ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
229 1 2 111 112 121 126 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑐 ) · 𝑒 ) = ( 𝑎 · ( 𝑐 · 𝑒 ) ) )
230 229 145 opeq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( 𝑎 · 𝑐 ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ = ⟨ ( 𝑎 · ( 𝑐 · 𝑒 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ )
231 230 eceq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( ( 𝑎 · 𝑐 ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ = [ ⟨ ( 𝑎 · ( 𝑐 · 𝑒 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
232 1 2 111 112 121 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑐 ) ∈ 𝐵 )
233 1 2 3 4 5 109 113 232 126 149 118 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( 𝑎 · 𝑐 ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ )
234 1 2 111 121 126 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑐 · 𝑒 ) ∈ 𝐵 )
235 1 2 3 4 5 109 113 112 234 122 152 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · ( 𝑐 · 𝑒 ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
236 231 233 235 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
237 1 2 3 4 5 109 113 112 121 122 115 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ )
238 237 oveq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
239 1 2 3 4 5 109 113 121 126 115 118 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ )
240 239 oveq2d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
241 236 238 240 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
242 160 161 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) )
243 242 163 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
244 161 163 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
245 160 244 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
246 241 243 245 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
247 246 170 r19.29vva ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
248 247 174 r19.29vva ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
249 248 177 r19.29vva ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
250 196 ad10antr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ∼ Er ( 𝐵 × 𝑆 ) )
251 eqidd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ = ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ )
252 1 2 111 112 123 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑏 ) ∈ 𝐵 )
253 1 3 2 111 252 141 133 ringdid ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑏 ) · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) = ( ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) + ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) ) )
254 1 2 111 112 123 151 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑏 ) · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) = ( 𝑎 · ( 𝑏 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
255 253 254 eqtr3d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) + ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) ) = ( 𝑎 · ( 𝑏 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
256 11 crngmgp ⊢ ( 𝑅 ∈ CRing → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
257 6 256 syl ⊢ ( 𝜑 → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
258 257 ad10antr ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( mulGrp ‘ 𝑅 ) ∈ CMnd )
259 12 95 258 112 121 123 119 cmn4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) = ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) )
260 12 95 258 112 126 123 116 cmn4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) = ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) )
261 259 260 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) = ( ( ( 𝑎 · 𝑏 ) · ( 𝑐 · 𝑓 ) ) + ( ( 𝑎 · 𝑏 ) · ( 𝑒 · 𝑑 ) ) ) )
262 1 2 109 123 112 151 crng12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) = ( 𝑎 · ( 𝑏 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
263 255 261 262 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) = ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
264 1 2 109 127 123 119 crng12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) = ( 𝑏 · ( ( 𝑏 · 𝑑 ) · 𝑓 ) ) )
265 145 oveq2d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( ( 𝑏 · 𝑑 ) · 𝑓 ) ) = ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) )
266 264 265 eqtrd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) = ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) )
267 263 266 opeq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ = ⟨ ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) , ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) ⟩ )
268 1 2 111 112 151 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ∈ 𝐵 )
269 1 2 111 123 268 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) ∈ 𝐵 )
270 95 113 122 152 submcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑑 · 𝑓 ) ) ∈ 𝑆 )
271 95 113 122 270 submcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) ∈ 𝑆 )
272 eqidd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) = ( 𝑏 · ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) ) )
273 eqidd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) = ( 𝑏 · ( 𝑏 · ( 𝑑 · 𝑓 ) ) ) )
274 1 5 109 113 2 251 267 268 269 270 271 122 272 273 erlbr2d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ∼ ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ )
275 250 274 erthi ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ = [ ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ ] ∼ )
276 1 2 3 4 5 109 113 112 151 122 152 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) ) , ( 𝑏 · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
277 1 2 111 112 126 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑎 · 𝑒 ) ∈ 𝐵 )
278 95 113 122 118 submcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑓 ) ∈ 𝑆 )
279 1 2 3 4 5 109 113 232 277 149 278 20 rlocaddval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( ( ( 𝑎 · 𝑐 ) · ( 𝑏 · 𝑓 ) ) + ( ( 𝑎 · 𝑒 ) · ( 𝑏 · 𝑑 ) ) ) , ( ( 𝑏 · 𝑑 ) · ( 𝑏 · 𝑓 ) ) ⟩ ] ∼ )
280 275 276 279 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ) )
281 157 oveq2d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( ( 𝑐 · 𝑓 ) + ( 𝑒 · 𝑑 ) ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
282 1 2 3 4 5 109 113 112 126 122 118 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ )
283 237 282 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ) )
284 280 281 283 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
285 160 165 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
286 160 163 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
287 242 286 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
288 284 285 287 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
289 288 170 r19.29vva ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
290 289 174 r19.29vva ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
291 290 177 r19.29vva ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( 𝑥 ( .r ‘ 𝐿 ) ( 𝑦 ( +g ‘ 𝐿 ) 𝑧 ) ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) ( +g ‘ 𝐿 ) ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ) )
292 1 3 2 111 117 124 126 ringdird ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) = ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) )
293 292 opeq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ = ⟨ ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ )
294 1 2 111 117 126 119 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) · 𝑓 ) = ( ( 𝑎 · 𝑑 ) · ( 𝑒 · 𝑓 ) ) )
295 1 2 111 117 126 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑑 ) · 𝑒 ) ∈ 𝐵 )
296 1 2 109 119 295 crngcomd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) = ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) · 𝑓 ) )
297 12 95 258 112 126 116 119 cmn4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) = ( ( 𝑎 · 𝑑 ) · ( 𝑒 · 𝑓 ) ) )
298 294 296 297 3eqtr4rd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) = ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) )
299 1 2 111 124 126 119 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑐 · 𝑏 ) · 𝑒 ) · 𝑓 ) = ( ( 𝑐 · 𝑏 ) · ( 𝑒 · 𝑓 ) ) )
300 1 2 111 124 126 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑏 ) · 𝑒 ) ∈ 𝐵 )
301 1 2 109 119 300 crngcomd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) = ( ( ( 𝑐 · 𝑏 ) · 𝑒 ) · 𝑓 ) )
302 12 95 258 121 126 123 119 cmn4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) = ( ( 𝑐 · 𝑏 ) · ( 𝑒 · 𝑓 ) ) )
303 299 301 302 3eqtr4rd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) = ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) )
304 298 303 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) = ( ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) + ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
305 1 3 2 111 119 295 300 ringdid ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) = ( ( 𝑓 · ( ( 𝑎 · 𝑑 ) · 𝑒 ) ) + ( 𝑓 · ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
306 304 305 eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) = ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
307 114 278 sseldd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑓 ) ∈ 𝐵 )
308 1 2 111 116 307 119 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑑 · ( 𝑏 · 𝑓 ) ) · 𝑓 ) = ( 𝑑 · ( ( 𝑏 · 𝑓 ) · 𝑓 ) ) )
309 1 2 109 123 116 crngcomd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) = ( 𝑑 · 𝑏 ) )
310 309 oveq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) = ( ( 𝑑 · 𝑏 ) · 𝑓 ) )
311 1 2 111 116 123 119 ringassd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑑 · 𝑏 ) · 𝑓 ) = ( 𝑑 · ( 𝑏 · 𝑓 ) ) )
312 310 311 eqtrd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) = ( 𝑑 · ( 𝑏 · 𝑓 ) ) )
313 312 oveq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) = ( ( 𝑑 · ( 𝑏 · 𝑓 ) ) · 𝑓 ) )
314 1 2 109 307 116 119 crng12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) = ( 𝑑 · ( ( 𝑏 · 𝑓 ) · 𝑓 ) ) )
315 308 313 314 3eqtr4rd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) = ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) )
316 306 315 opeq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ = ⟨ ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) , ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) ⟩ )
317 1 3 110 295 300 grpcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ∈ 𝐵 )
318 1 2 111 119 317 ringcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) ∈ 𝐵 )
319 145 270 eqeltrd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) ∈ 𝑆 )
320 95 113 319 118 submcld ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) ∈ 𝑆 )
321 eqidd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) = ( 𝑓 · ( ( ( 𝑎 · 𝑑 ) · 𝑒 ) + ( ( 𝑐 · 𝑏 ) · 𝑒 ) ) ) )
322 114 319 sseldd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑏 · 𝑑 ) · 𝑓 ) ∈ 𝐵 )
323 1 2 109 322 119 crngcomd ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( ( 𝑏 · 𝑑 ) · 𝑓 ) · 𝑓 ) = ( 𝑓 · ( ( 𝑏 · 𝑑 ) · 𝑓 ) ) )
324 1 5 109 113 2 293 316 317 318 319 320 118 321 323 erlbr2d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ∼ ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ )
325 250 324 erthi ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → [ ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ = [ ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
326 1 2 3 4 5 109 113 148 126 149 118 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = [ ⟨ ( ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) · 𝑒 ) , ( ( 𝑏 · 𝑑 ) · 𝑓 ) ⟩ ] ∼ )
327 1 2 3 4 5 109 113 277 234 278 152 20 rlocaddval ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) = [ ⟨ ( ( ( 𝑎 · 𝑒 ) · ( 𝑑 · 𝑓 ) ) + ( ( 𝑐 · 𝑒 ) · ( 𝑏 · 𝑓 ) ) ) , ( ( 𝑏 · 𝑓 ) · ( 𝑑 · 𝑓 ) ) ⟩ ] ∼ )
328 325 326 327 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
329 155 oveq1d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( [ ⟨ ( ( 𝑎 · 𝑑 ) + ( 𝑐 · 𝑏 ) ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
330 282 239 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) = ( [ ⟨ ( 𝑎 · 𝑒 ) , ( 𝑏 · 𝑓 ) ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ ( 𝑐 · 𝑒 ) , ( 𝑑 · 𝑓 ) ⟩ ] ∼ ) )
331 328 329 330 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
332 162 163 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( +g ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) )
333 286 244 oveq12d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) = ( ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ( +g ‘ 𝐿 ) ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) ) )
334 331 332 333 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) ∧ 𝑒 ∈ 𝐵 ) ∧ 𝑓 ∈ 𝑆 ) ∧ 𝑧 = [ ⟨ 𝑒 , 𝑓 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
335 334 170 r19.29vva ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
336 335 174 r19.29vva ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
337 336 177 r19.29vva ⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑧 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ) → ( ( 𝑥 ( +g ‘ 𝐿 ) 𝑦 ) ( .r ‘ 𝐿 ) 𝑧 ) = ( ( 𝑥 ( .r ‘ 𝐿 ) 𝑧 ) ( +g ‘ 𝐿 ) ( 𝑦 ( .r ‘ 𝐿 ) 𝑧 ) ) )
338 14 27 sseldd ⊢ ( 𝜑 → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
339 338 27 opelxpd ⊢ ( 𝜑 → ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) )
340 29 ecelqsi ⊢ ( ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ∈ ( 𝐵 × 𝑆 ) → [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
341 339 340 syl ⊢ ( 𝜑 → [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) )
342 35 oveq2d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
343 1 2 24 37 43 ringlidmd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( ( 1r ‘ 𝑅 ) · 𝑎 ) = 𝑎 )
344 343 50 opeq12d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ ( ( 1r ‘ 𝑅 ) · 𝑎 ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ = ⟨ 𝑎 , 𝑏 ⟩ )
345 344 eceq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ ( ( 1r ‘ 𝑅 ) · 𝑎 ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
346 39 55 sseldd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 1r ‘ 𝑅 ) ∈ 𝐵 )
347 1 2 3 4 5 53 38 346 43 55 40 218 rlocmulval ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( ( 1r ‘ 𝑅 ) · 𝑎 ) , ( ( 1r ‘ 𝑅 ) · 𝑏 ) ⟩ ] ∼ )
348 345 347 35 3eqtr4d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = 𝑥 )
349 342 348 eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) 𝑥 ) = 𝑥 )
350 349 60 r19.29vva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ( .r ‘ 𝐿 ) 𝑥 ) = 𝑥 )
351 1 2 3 4 5 53 38 43 346 40 55 218 rlocmulval ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = [ ⟨ ( 𝑎 · ( 1r ‘ 𝑅 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
352 35 oveq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) )
353 44 eqcomd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → 𝑎 = ( 𝑎 · ( 1r ‘ 𝑅 ) ) )
354 353 69 opeq12d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ⟨ 𝑎 , 𝑏 ⟩ = ⟨ ( 𝑎 · ( 1r ‘ 𝑅 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ )
355 354 eceq1d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ = [ ⟨ ( 𝑎 · ( 1r ‘ 𝑅 ) ) , ( 𝑏 · ( 1r ‘ 𝑅 ) ) ⟩ ] ∼ )
356 351 352 355 3eqtr4d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ )
357 356 35 eqtr4d ⊢ ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
358 357 60 r19.29vva ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) [ ⟨ ( 1r ‘ 𝑅 ) , ( 1r ‘ 𝑅 ) ⟩ ] ∼ ) = 𝑥 )
359 1 2 80 82 83 crngcomd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑎 · 𝑐 ) = ( 𝑐 · 𝑎 ) )
360 1 2 80 92 90 crngcomd ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑏 · 𝑑 ) = ( 𝑑 · 𝑏 ) )
361 359 360 opeq12d ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ = ⟨ ( 𝑐 · 𝑎 ) , ( 𝑑 · 𝑏 ) ⟩ )
362 361 eceq1d ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → [ ⟨ ( 𝑎 · 𝑐 ) , ( 𝑏 · 𝑑 ) ⟩ ] ∼ = [ ⟨ ( 𝑐 · 𝑎 ) , ( 𝑑 · 𝑏 ) ⟩ ] ∼ )
363 1 2 3 4 5 80 81 83 82 85 84 218 rlocmulval ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) = [ ⟨ ( 𝑐 · 𝑎 ) , ( 𝑑 · 𝑏 ) ⟩ ] ∼ )
364 362 219 363 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
365 78 77 oveq12d ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) = ( [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ( .r ‘ 𝐿 ) [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) )
366 364 217 365 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) ∧ 𝑐 ∈ 𝐵 ) ∧ 𝑑 ∈ 𝑆 ) ∧ 𝑦 = [ ⟨ 𝑐 , 𝑑 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
367 366 103 r19.29vva ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑎 ∈ 𝐵 ) ∧ 𝑏 ∈ 𝑆 ) ∧ 𝑥 = [ ⟨ 𝑎 , 𝑏 ⟩ ] ∼ ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
368 367 106 r19.29vva ⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
369 368 3impa ⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ∧ 𝑦 ∈ ( ( 𝐵 × 𝑆 ) / ∼ ) ) → ( 𝑥 ( .r ‘ 𝐿 ) 𝑦 ) = ( 𝑦 ( .r ‘ 𝐿 ) 𝑥 ) )
370 15 16 17 216 228 249 291 337 341 350 358 369 iscrngd ⊢ ( 𝜑 → 𝐿 ∈ CRing )