Metamath Proof Explorer

Theorem cbvproddavw2

Description: Change bound variable and the set of integers in a product. Deduction form. (Contributed by GG, 14-Aug-2025)

Ref Expression
Hypotheses cbvproddavw2.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
cbvproddavw2.2 ⊢ ( ( 𝜑 ∧ 𝑗 = 𝑘 ) → 𝐶 = 𝐷 )
Assertion cbvproddavw2 ( 𝜑 → ∏ 𝑗 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐷 )

Proof

Step Hyp Ref Expression
1 cbvproddavw2.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 cbvproddavw2.2 ⊢ ( ( 𝜑 ∧ 𝑗 = 𝑘 ) → 𝐶 = 𝐷 )
3 1 sseq1d ⊢ ( 𝜑 → ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ↔ 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ) )
4 simpr ⊢ ( ( 𝜑 ∧ 𝑗 = 𝑘 ) → 𝑗 = 𝑘 )
5 1 adantr ⊢ ( ( 𝜑 ∧ 𝑗 = 𝑘 ) → 𝐴 = 𝐵 )
6 4 5 eleq12d ⊢ ( ( 𝜑 ∧ 𝑗 = 𝑘 ) → ( 𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐵 ) )
7 6 2 ifbieq1d ⊢ ( ( 𝜑 ∧ 𝑗 = 𝑘 ) → if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) = if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) )
8 7 cbvmptdavw ⊢ ( 𝜑 → ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) )
9 8 seqeq3d ⊢ ( 𝜑 → seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) = seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) )
10 9 breq1d ⊢ ( 𝜑 → ( seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ↔ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) )
11 10 anbi2d ⊢ ( 𝜑 → ( ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ↔ ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ) )
12 11 exbidv ⊢ ( 𝜑 → ( ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ↔ ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ) )
13 12 rexbidv ⊢ ( 𝜑 → ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ) )
14 8 seqeq3d ⊢ ( 𝜑 → seq 𝑚 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) = seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) )
15 14 breq1d ⊢ ( 𝜑 → ( seq 𝑚 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ↔ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑥 ) )
16 3 13 15 3anbi123d ⊢ ( 𝜑 → ( ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ↔ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑥 ) ) )
17 16 rexbidv ⊢ ( 𝜑 → ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ↔ ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑥 ) ) )
18 1 f1oeq3d ⊢ ( 𝜑 → ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ↔ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ) )
19 2 cbvcsbdavw ⊢ ( 𝜑 → ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 = ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 )
20 19 mpteq2dv ⊢ ( 𝜑 → ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) = ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) )
21 20 seqeq3d ⊢ ( 𝜑 → seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) = seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) )
22 21 fveq1d ⊢ ( 𝜑 → ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) )
23 22 eqeq2d ⊢ ( 𝜑 → ( 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) ↔ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) ) )
24 18 23 anbi12d ⊢ ( 𝜑 → ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) ) ) )
25 24 exbidv ⊢ ( 𝜑 → ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) ) ) )
26 25 rexbidv ⊢ ( 𝜑 → ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) ) ) )
27 17 26 orbi12d ⊢ ( 𝜑 → ( ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) ↔ ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) ) ) ) )
28 27 iotabidv ⊢ ( 𝜑 → ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) ) = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) ) ) ) )
29 df-prod ⊢ ∏ 𝑗 ∈ 𝐴 𝐶 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑗 ∈ ℤ ↦ if ( 𝑗 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑗 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
30 df-prod ⊢ ∏ 𝑘 ∈ 𝐵 𝐷 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐷 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐷 ) ) ‘ 𝑚 ) ) ) )
31 28 29 30 3eqtr4g ⊢ ( 𝜑 → ∏ 𝑗 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐷 )