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Metamath Proof Explorer

Theorem clmmulg

Description: The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015)

Ref Expression
Hypotheses clmmulg.1 ⊢ 𝑉 = ( Base ‘ 𝑊 )
clmmulg.2 ⊢ ∙ = ( .g ‘ 𝑊 )
clmmulg.3 ⊢ · = ( ·𝑠 ‘ 𝑊 )
Assertion clmmulg ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 ∙ 𝐵 ) = ( 𝐴 · 𝐵 ) )

Proof

Step Hyp Ref Expression
1 clmmulg.1 ⊢ 𝑉 = ( Base ‘ 𝑊 )
2 clmmulg.2 ⊢ ∙ = ( .g ‘ 𝑊 )
3 clmmulg.3 ⊢ · = ( ·𝑠 ‘ 𝑊 )
4 oveq1 ⊢ ( 𝑥 = 0 → ( 𝑥 ∙ 𝐵 ) = ( 0 ∙ 𝐵 ) )
5 oveq1 ⊢ ( 𝑥 = 0 → ( 𝑥 · 𝐵 ) = ( 0 · 𝐵 ) )
6 4 5 eqeq12d ⊢ ( 𝑥 = 0 → ( ( 𝑥 ∙ 𝐵 ) = ( 𝑥 · 𝐵 ) ↔ ( 0 ∙ 𝐵 ) = ( 0 · 𝐵 ) ) )
7 oveq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∙ 𝐵 ) = ( 𝑦 ∙ 𝐵 ) )
8 oveq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 · 𝐵 ) = ( 𝑦 · 𝐵 ) )
9 7 8 eqeq12d ⊢ ( 𝑥 = 𝑦 → ( ( 𝑥 ∙ 𝐵 ) = ( 𝑥 · 𝐵 ) ↔ ( 𝑦 ∙ 𝐵 ) = ( 𝑦 · 𝐵 ) ) )
10 oveq1 ⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 ∙ 𝐵 ) = ( ( 𝑦 + 1 ) ∙ 𝐵 ) )
11 oveq1 ⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( 𝑥 · 𝐵 ) = ( ( 𝑦 + 1 ) · 𝐵 ) )
12 10 11 eqeq12d ⊢ ( 𝑥 = ( 𝑦 + 1 ) → ( ( 𝑥 ∙ 𝐵 ) = ( 𝑥 · 𝐵 ) ↔ ( ( 𝑦 + 1 ) ∙ 𝐵 ) = ( ( 𝑦 + 1 ) · 𝐵 ) ) )
13 oveq1 ⊢ ( 𝑥 = - 𝑦 → ( 𝑥 ∙ 𝐵 ) = ( - 𝑦 ∙ 𝐵 ) )
14 oveq1 ⊢ ( 𝑥 = - 𝑦 → ( 𝑥 · 𝐵 ) = ( - 𝑦 · 𝐵 ) )
15 13 14 eqeq12d ⊢ ( 𝑥 = - 𝑦 → ( ( 𝑥 ∙ 𝐵 ) = ( 𝑥 · 𝐵 ) ↔ ( - 𝑦 ∙ 𝐵 ) = ( - 𝑦 · 𝐵 ) ) )
16 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∙ 𝐵 ) = ( 𝐴 ∙ 𝐵 ) )
17 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 · 𝐵 ) = ( 𝐴 · 𝐵 ) )
18 16 17 eqeq12d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∙ 𝐵 ) = ( 𝑥 · 𝐵 ) ↔ ( 𝐴 ∙ 𝐵 ) = ( 𝐴 · 𝐵 ) ) )
19 eqid ⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
20 1 19 2 mulg0 ⊢ ( 𝐵 ∈ 𝑉 → ( 0 ∙ 𝐵 ) = ( 0g ‘ 𝑊 ) )
21 20 adantl ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( 0 ∙ 𝐵 ) = ( 0g ‘ 𝑊 ) )
22 eqid ⊢ ( Scalar ‘ 𝑊 ) = ( Scalar ‘ 𝑊 )
23 1 22 3 19 clm0vs ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( 0 · 𝐵 ) = ( 0g ‘ 𝑊 ) )
24 21 23 eqtr4d ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( 0 ∙ 𝐵 ) = ( 0 · 𝐵 ) )
25 oveq1 ⊢ ( ( 𝑦 ∙ 𝐵 ) = ( 𝑦 · 𝐵 ) → ( ( 𝑦 ∙ 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) = ( ( 𝑦 · 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) )
26 clmgrp ⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ Grp )
27 26 grpmndd ⊢ ( 𝑊 ∈ ℂMod → 𝑊 ∈ Mnd )
28 27 ad2antrr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑊 ∈ Mnd )
29 simpr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑦 ∈ ℕ0 )
30 simplr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 𝐵 ∈ 𝑉 )
31 eqid ⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
32 1 2 31 mulgnn0p1 ⊢ ( ( 𝑊 ∈ Mnd ∧ 𝑦 ∈ ℕ0 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝑦 + 1 ) ∙ 𝐵 ) = ( ( 𝑦 ∙ 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) )
33 28 29 30 32 syl3anc ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑦 + 1 ) ∙ 𝐵 ) = ( ( 𝑦 ∙ 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) )
34 simpll ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑊 ∈ ℂMod )
35 eqid ⊢ ( Base ‘ ( Scalar ‘ 𝑊 ) ) = ( Base ‘ ( Scalar ‘ 𝑊 ) )
36 22 35 clmzss ⊢ ( 𝑊 ∈ ℂMod → ℤ ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
37 36 ad2antrr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ℤ ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
38 nn0z ⊢ ( 𝑦 ∈ ℕ0 → 𝑦 ∈ ℤ )
39 38 adantl ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑦 ∈ ℤ )
40 37 39 sseldd ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 𝑦 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
41 1zzd ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 1 ∈ ℤ )
42 37 41 sseldd ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
43 1 22 3 35 31 clmvsdir ⊢ ( ( 𝑊 ∈ ℂMod ∧ ( 𝑦 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 1 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) ∧ 𝐵 ∈ 𝑉 ) ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( ( 𝑦 · 𝐵 ) ( +g ‘ 𝑊 ) ( 1 · 𝐵 ) ) )
44 34 40 42 30 43 syl13anc ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( ( 𝑦 · 𝐵 ) ( +g ‘ 𝑊 ) ( 1 · 𝐵 ) ) )
45 1 3 clmvs1 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( 1 · 𝐵 ) = 𝐵 )
46 45 adantr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ( 1 · 𝐵 ) = 𝐵 )
47 46 oveq2d ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑦 · 𝐵 ) ( +g ‘ 𝑊 ) ( 1 · 𝐵 ) ) = ( ( 𝑦 · 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) )
48 44 47 eqtrd ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑦 + 1 ) · 𝐵 ) = ( ( 𝑦 · 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) )
49 33 48 eqeq12d ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( ( 𝑦 + 1 ) ∙ 𝐵 ) = ( ( 𝑦 + 1 ) · 𝐵 ) ↔ ( ( 𝑦 ∙ 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) = ( ( 𝑦 · 𝐵 ) ( +g ‘ 𝑊 ) 𝐵 ) ) )
50 25 49 imbitrrid ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ0 ) → ( ( 𝑦 ∙ 𝐵 ) = ( 𝑦 · 𝐵 ) → ( ( 𝑦 + 1 ) ∙ 𝐵 ) = ( ( 𝑦 + 1 ) · 𝐵 ) ) )
51 50 ex ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( 𝑦 ∈ ℕ0 → ( ( 𝑦 ∙ 𝐵 ) = ( 𝑦 · 𝐵 ) → ( ( 𝑦 + 1 ) ∙ 𝐵 ) = ( ( 𝑦 + 1 ) · 𝐵 ) ) ) )
52 fveq2 ⊢ ( ( 𝑦 ∙ 𝐵 ) = ( 𝑦 · 𝐵 ) → ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 ∙ 𝐵 ) ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 · 𝐵 ) ) )
53 26 ad2antrr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → 𝑊 ∈ Grp )
54 nnz ⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℤ )
55 54 adantl ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → 𝑦 ∈ ℤ )
56 simplr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → 𝐵 ∈ 𝑉 )
57 eqid ⊢ ( invg ‘ 𝑊 ) = ( invg ‘ 𝑊 )
58 1 2 57 mulgneg ⊢ ( ( 𝑊 ∈ Grp ∧ 𝑦 ∈ ℤ ∧ 𝐵 ∈ 𝑉 ) → ( - 𝑦 ∙ 𝐵 ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 ∙ 𝐵 ) ) )
59 53 55 56 58 syl3anc ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → ( - 𝑦 ∙ 𝐵 ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 ∙ 𝐵 ) ) )
60 simpll ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → 𝑊 ∈ ℂMod )
61 36 ad2antrr ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → ℤ ⊆ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
62 61 55 sseldd ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → 𝑦 ∈ ( Base ‘ ( Scalar ‘ 𝑊 ) ) )
63 1 22 3 57 35 60 56 62 clmvsneg ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 · 𝐵 ) ) = ( - 𝑦 · 𝐵 ) )
64 63 eqcomd ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → ( - 𝑦 · 𝐵 ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 · 𝐵 ) ) )
65 59 64 eqeq12d ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → ( ( - 𝑦 ∙ 𝐵 ) = ( - 𝑦 · 𝐵 ) ↔ ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 ∙ 𝐵 ) ) = ( ( invg ‘ 𝑊 ) ‘ ( 𝑦 · 𝐵 ) ) ) )
66 52 65 imbitrrid ⊢ ( ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) ∧ 𝑦 ∈ ℕ ) → ( ( 𝑦 ∙ 𝐵 ) = ( 𝑦 · 𝐵 ) → ( - 𝑦 ∙ 𝐵 ) = ( - 𝑦 · 𝐵 ) ) )
67 66 ex ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( 𝑦 ∈ ℕ → ( ( 𝑦 ∙ 𝐵 ) = ( 𝑦 · 𝐵 ) → ( - 𝑦 ∙ 𝐵 ) = ( - 𝑦 · 𝐵 ) ) ) )
68 6 9 12 15 18 24 51 67 zindd ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 ∈ ℤ → ( 𝐴 ∙ 𝐵 ) = ( 𝐴 · 𝐵 ) ) )
69 68 3impia ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐵 ∈ 𝑉 ∧ 𝐴 ∈ ℤ ) → ( 𝐴 ∙ 𝐵 ) = ( 𝐴 · 𝐵 ) )
70 69 3com23 ⊢ ( ( 𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 ∙ 𝐵 ) = ( 𝐴 · 𝐵 ) )