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Metamath Proof Explorer

Theorem prodeq1i

Description: Equality inference for product. (Contributed by Scott Fenton, 4-Dec-2017) Remove DV conditions. (Revised by GG, 1-Sep-2025)

Ref Expression
Hypothesis prodeq1i.1 ⊢ 𝐴 = 𝐵
Assertion prodeq1i ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐶

Proof

Step Hyp Ref Expression
1 prodeq1i.1 ⊢ 𝐴 = 𝐵
2 1 sseq1i ⊢ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ↔ 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) )
3 1 eleq2i ⊢ ( 𝑘 ∈ 𝐴 ↔ 𝑘 ∈ 𝐵 )
4 ifbi ⊢ ( ( 𝑘 ∈ 𝐴 ↔ 𝑘 ∈ 𝐵 ) → if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) )
5 3 4 ax-mp ⊢ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) = if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 )
6 5 mpteq2i ⊢ ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) )
7 seqeq3 ⊢ ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) → seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) = seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) )
8 6 7 ax-mp ⊢ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) = seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) )
9 8 breq1i ⊢ ( seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ↔ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 )
10 9 anbi2i ⊢ ( ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ↔ ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) )
11 10 exbii ⊢ ( ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ↔ ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) )
12 11 rexbii ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) )
13 seqeq3 ⊢ ( ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) = ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) → seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) = seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) )
14 6 13 ax-mp ⊢ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) = seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) )
15 14 breq1i ⊢ ( seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐴 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ↔ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑥 )
16 2 12 15 3anbi123i ⊢ ( ( 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 ) ↔ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) )
17 16 rexbii ⊢ ( ∃ 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 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) )
18 f1oeq3 ⊢ ( 𝐴 = 𝐵 → ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ↔ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ) )
19 1 18 ax-mp ⊢ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ↔ 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 )
20 19 anbi1i ⊢ ( ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) )
21 20 exbii ⊢ ( ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) )
22 21 rexbii ⊢ ( ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ↔ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) )
23 17 22 orbi12i ⊢ ( ( ∃ 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 ) ) ) ↔ ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
24 23 iotabii ⊢ ( ℩ 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 ) ) ) ) = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
25 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 ) ) ) )
26 df-prod ⊢ ∏ 𝑘 ∈ 𝐵 𝐶 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐵 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑚 ) ∃ 𝑦 ( 𝑦 ≠ 0 ∧ seq 𝑛 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑦 ) ∧ seq 𝑚 ( · , ( 𝑘 ∈ ℤ ↦ if ( 𝑘 ∈ 𝐵 , 𝐶 , 1 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐵 ∧ 𝑥 = ( seq 1 ( · , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐶 ) ) ‘ 𝑚 ) ) ) )
27 24 25 26 3eqtr4i ⊢ ∏ 𝑘 ∈ 𝐴 𝐶 = ∏ 𝑘 ∈ 𝐵 𝐶