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Metamath Proof Explorer

Theorem prodeq2ii

Description: Equality theorem for product, with the class expressions B and C guarded by _I to be always sets. (Contributed by Scott Fenton, 4-Dec-2017)

Ref Expression
Assertion prodeq2ii ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )

Proof

Step Hyp Ref Expression
1 eluzelz ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) → 𝑛 ∈ ℤ )
2 1 adantl ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → 𝑛 ∈ ℤ )
3 nfra1 ⊢ Ⅎ 𝑘 ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 )
4 rsp ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ( 𝑘 ∈ 𝐴 → ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ) )
5 4 adantr ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑘 ∈ ℤ ) → ( 𝑘 ∈ 𝐴 → ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ) )
6 ifeq1 ⊢ ( ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐵 ) , ( I ‘ 1 ) ) = if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐶 ) , ( I ‘ 1 ) ) )
7 5 6 syl6 ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑘 ∈ ℤ ) → ( 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐵 ) , ( I ‘ 1 ) ) = if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐶 ) , ( I ‘ 1 ) ) ) )
8 iffalse ⊢ ( ¬ 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐵 ) , ( I ‘ 1 ) ) = ( I ‘ 1 ) )
9 iffalse ⊢ ( ¬ 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐶 ) , ( I ‘ 1 ) ) = ( I ‘ 1 ) )
10 8 9 eqtr4d ⊢ ( ¬ 𝑘 ∈ 𝐴 → if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐵 ) , ( I ‘ 1 ) ) = if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐶 ) , ( I ‘ 1 ) ) )
11 7 10 pm2.61d1 ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑘 ∈ ℤ ) → if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐵 ) , ( I ‘ 1 ) ) = if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐶 ) , ( I ‘ 1 ) ) )
12 fvif ⊢ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) = if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐵 ) , ( I ‘ 1 ) )
13 fvif ⊢ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) = if ( 𝑘 ∈ 𝐴 , ( I ‘ 𝐶 ) , ( I ‘ 1 ) )
14 11 12 13 3eqtr4g ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑘 ∈ ℤ ) → ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) = ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) )
15 3 14 mpteq2da ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) = ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) )
16 15 adantr ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) = ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) )
17 16 fveq1d ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ‘ 𝑥 ) )
18 17 adantlr ⊢ ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ‘ 𝑥 ) )
19 eqid ⊢ ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) )
20 eqid ⊢ ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) = ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) )
21 19 20 fvmptex ⊢ ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ‘ 𝑥 )
22 eqid ⊢ ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) )
23 eqid ⊢ ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) = ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) )
24 22 23 fvmptex ⊢ ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ‘ 𝑥 )
25 18 21 24 3eqtr4g ⊢ ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑛 ) ) → ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ‘ 𝑥 ) )
26 2 25 seqfeq ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) = seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) )
27 26 breq1d ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ↔ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) )
28 27 anbi2d ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ↔ ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ) )
29 28 exbidv ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ↔ ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ) )
30 29 rexbidva ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ) )
31 30 adantr ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℤ ) → ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ) )
32 simpr ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℤ ) → 𝑚 ∈ ℤ )
33 15 adantr ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) = ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) )
34 33 fveq1d ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ ( I ‘ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ‘ 𝑥 ) )
35 34 21 24 3eqtr4g ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ‘ 𝑥 ) )
36 35 adantlr ⊢ ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℤ ) ∧ 𝑥 ∈ ( ℤ≥ ‘ 𝑚 ) ) → ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ‘ 𝑥 ) = ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ‘ 𝑥 ) )
37 32 36 seqfeq ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℤ ) → seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) = seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) )
38 37 breq1d ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℤ ) → ( seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑥 ↔ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) )
39 31 38 3anbi23d ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℤ ) → ( ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑥 ) ↔ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ) )
40 39 rexbidva ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑥 ) ↔ ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ) )
41 simplr ⊢ ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → 𝑚 ∈ ℕ )
42 nnuz ⊢ ℕ = ( ℤ≥ ‘ 1 )
43 41 42 eleqtrdi ⊢ ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → 𝑚 ∈ ( ℤ≥ ‘ 1 ) )
44 f1of ⊢ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 → 𝑓 : ( 1 ... 𝑚 ) ⟶ 𝐴 )
45 44 ad2antlr ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → 𝑓 : ( 1 ... 𝑚 ) ⟶ 𝐴 )
46 ffvelcdm ⊢ ( ( 𝑓 : ( 1 ... 𝑚 ) ⟶ 𝐴 ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ( 𝑓 ‘ 𝑥 ) ∈ 𝐴 )
47 45 46 sylancom ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ( 𝑓 ‘ 𝑥 ) ∈ 𝐴 )
48 simplll ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) )
49 nfcsb1v ⊢ Ⅎ 𝑘 ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 )
50 nfcsb1v ⊢ Ⅎ 𝑘 ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 )
51 49 50 nfeq ⊢ Ⅎ 𝑘 ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 ) = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 )
52 csbeq1a ⊢ ( 𝑘 = ( 𝑓 ‘ 𝑥 ) → ( I ‘ 𝐵 ) = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 ) )
53 csbeq1a ⊢ ( 𝑘 = ( 𝑓 ‘ 𝑥 ) → ( I ‘ 𝐶 ) = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 ) )
54 52 53 eqeq12d ⊢ ( 𝑘 = ( 𝑓 ‘ 𝑥 ) → ( ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ↔ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 ) = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 ) ) )
55 51 54 rspc ⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ 𝐴 → ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 ) = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 ) ) )
56 47 48 55 sylc ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 ) = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 ) )
57 fvex ⊢ ( 𝑓 ‘ 𝑥 ) ∈ V
58 csbfv2g ⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ V → ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐵 ) )
59 57 58 ax-mp ⊢ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐵 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐵 )
60 csbfv2g ⊢ ( ( 𝑓 ‘ 𝑥 ) ∈ V → ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐶 ) )
61 57 60 ax-mp ⊢ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ ( I ‘ 𝐶 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐶 )
62 56 59 61 3eqtr3g ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐵 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐶 ) )
63 elfznn ⊢ ( 𝑥 ∈ ( 1 ... 𝑚 ) → 𝑥 ∈ ℕ )
64 63 adantl ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → 𝑥 ∈ ℕ )
65 fveq2 ⊢ ( 𝑛 = 𝑥 → ( 𝑓 ‘ 𝑛 ) = ( 𝑓 ‘ 𝑥 ) )
66 65 csbeq1d ⊢ ( 𝑛 = 𝑥 → ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐵 )
67 eqid ⊢ ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) = ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 )
68 66 67 fvmpti ⊢ ( 𝑥 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ‘ 𝑥 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐵 ) )
69 64 68 syl ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ( ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ‘ 𝑥 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐵 ) )
70 65 csbeq1d ⊢ ( 𝑛 = 𝑥 → ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 = ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐶 )
71 eqid ⊢ ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) = ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 )
72 70 71 fvmpti ⊢ ( 𝑥 ∈ ℕ → ( ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ‘ 𝑥 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐶 ) )
73 64 72 syl ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ( ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ‘ 𝑥 ) = ( I ‘ ⦋ ( 𝑓 ‘ 𝑥 ) / 𝑘 ⦌ 𝐶 ) )
74 62 69 73 3eqtr4d ⊢ ( ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) ∧ 𝑥 ∈ ( 1 ... 𝑚 ) ) → ( ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ‘ 𝑥 ) = ( ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ‘ 𝑥 ) )
75 43 74 seqfveq ⊢ ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) )
76 75 eqeq2d ⊢ ( ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) ∧ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ) → ( 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ↔ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) )
77 76 pm5.32da ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) → ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ↔ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
78 77 exbidv ⊢ ( ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) ∧ 𝑚 ∈ ℕ ) → ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
79 78 rexbidva ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
80 40 79 orbi12d ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ↔ ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) ) )
81 80 iotabidv ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ) = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) ) )
82 df-prod ⊢ ∏ 𝑘 ∈ 𝐴 𝐵 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐵 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) )
83 df-prod ⊢ ∏ 𝑘 ∈ 𝐴 𝐶 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
84 81 82 83 3eqtr4g ⊢ ( ∀ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 ) = ( I ‘ 𝐶 ) → ∏ 𝑘 ∈ 𝐴 𝐵 = ∏ 𝑘 ∈ 𝐴 𝐶 )