Theorem cbvprod 15724

Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypotheses

Ref	Expression
cbvprod.1	⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
cbvprod.2	⊢ Ⅎ𝑘𝐴
cbvprod.3	⊢ Ⅎ𝑗𝐴
cbvprod.4	⊢ Ⅎ𝑘𝐵
cbvprod.5	⊢ Ⅎ𝑗𝐶

Assertion

Ref	Expression
cbvprod	⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Distinct variable group:   𝑗,𝑘

Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗,𝑘)

Proof of Theorem cbvprod

Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		biid 261	. . . . . 6 ⊢ (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑚))
2		cbvprod.2	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘𝐴
3	2	nfcri 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘 𝑗 ∈ 𝐴
4		cbvprod.4	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘𝐵
5		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘1
6	3, 4, 5	nfif 4507	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘if(𝑗 ∈ 𝐴, 𝐵, 1)
7		cbvprod.3	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑗𝐴
8	7	nfcri 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗 𝑘 ∈ 𝐴
9		cbvprod.5	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗𝐶
10		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗1
11	8, 9, 10	nfif 4507	. . . . . . . . . . . 12 ⊢ Ⅎ𝑗if(𝑘 ∈ 𝐴, 𝐶, 1)
12		eleq1w 2820	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
13		cbvprod.1	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑘 → 𝐵 = 𝐶)
14	12, 13	ifbieq1d 4501	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑘 → if(𝑗 ∈ 𝐴, 𝐵, 1) = if(𝑘 ∈ 𝐴, 𝐶, 1))
15	6, 11, 14	cbvmpt 5207	. . . . . . . . . . 11 ⊢ (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))
16		seqeq3 13831	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)) → seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
17	15, 16	ax-mp 5	. . . . . . . . . 10 ⊢ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)))
18	17	breq1i 5103	. . . . . . . . 9 ⊢ (seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)
19	18	anbi2i 624	. . . . . . . 8 ⊢ ((𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
20	19	exbii 1850	. . . . . . 7 ⊢ (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
21	20	rexbii 3094	. . . . . 6 ⊢ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
22		seqeq3 13831	. . . . . . . 8 ⊢ ((𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)) → seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
23	15, 22	ax-mp 5	. . . . . . 7 ⊢ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)))
24	23	breq1i 5103	. . . . . 6 ⊢ (seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)
25	1, 21, 24	3anbi123i 1155	. . . . 5 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥))
26	25	rexbii 3094	. . . 4 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥))
27	4, 9, 13	cbvcsbw 3856	. . . . . . . . . . 11 ⊢ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶
28	27	mpteq2i 5201	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
29		seqeq3 13831	. . . . . . . . . 10 ⊢ ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶) → seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵)) = seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)))
30	28, 29	ax-mp 5	. . . . . . . . 9 ⊢ seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵)) = seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
31	30	fveq1i 6830	. . . . . . . 8 ⊢ (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)
32	31	eqeq2i 2750	. . . . . . 7 ⊢ (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))
33	32	anbi2i 624	. . . . . 6 ⊢ ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
34	33	exbii 1850	. . . . 5 ⊢ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
35	34	rexbii 3094	. . . 4 ⊢ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
36	26, 35	orbi12i 913	. . 3 ⊢ ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
37	36	iotabii 6468	. 2 ⊢ (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
38		df-prod 15715	. 2 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐵))‘𝑚))))
39		df-prod 15715	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
40	37, 38, 39	3eqtr4i 2775	1 ⊢ ∏𝑗 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶

Syntax hints:   → wi 4   ∧ wa 397   ∨ wo 845   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Ⅎwnfc 2885   ≠ wne 2941  ∃wrex 3071  ⦋csb 3846   ⊆ wss 3901  ifcif 4477   class class class wbr 5096   ↦ cmpt 5179  ℩cio 6433  –1-1-onto→wf1o 6482  ‘cfv 6483  (class class class)co 7341  0cc0 10976  1c1 10977   · cmul 10981  ℕcn 12078  ℤcz 12424  ℤ_≥cuz 12687  ...cfz 13344  seqcseq 13826   ⇝ cli 15292  ∏cprod 15714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-xp 5630  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-iota 6435  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-seq 13827  df-prod 15715

This theorem is referenced by:  cbvprodv  15725  cbvprodi  15726  vonn0ioo2  44617  vonn0icc2  44619