Theorem cbvproddavw2 36865

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

Hypotheses

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

Assertion

Ref	Expression
cbvproddavw2	⊢ (𝜑 → ∏𝑗 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷)

Distinct variable groups:   𝜑,𝑗,𝑘   𝐴,𝑘   𝐵,𝑗   𝐶,𝑘   𝐷,𝑗

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

Proof of Theorem cbvproddavw2

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

Step	Hyp	Ref	Expression
1		cbvproddavw2.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	sseq1d 3969	. . . . . 6 ⊢ (𝜑 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐵 ⊆ (ℤ≥‘𝑚)))
3		simpr 490	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = 𝑘) → 𝑗 = 𝑘)
4	1	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑗 = 𝑘) → 𝐴 = 𝐵)
5	3, 4	eleq12d 2859	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑘) → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐵))
6		cbvproddavw2.2	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑗 = 𝑘) → 𝐶 = 𝐷)
7	5, 6	ifbieq1d 4514	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑗 = 𝑘) → if(𝑗 ∈ 𝐴, 𝐶, 1) = if(𝑘 ∈ 𝐵, 𝐷, 1))
8	7	cbvmptdavw 36836	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1)))
9	8	seqeq3d 14063	. . . . . . . . . 10 ⊢ (𝜑 → seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))))
10	9	breq1d 5121	. . . . . . . . 9 ⊢ (𝜑 → (seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦))
11	10	anbi2d 642	. . . . . . . 8 ⊢ (𝜑 → ((𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦)))
12	11	exbidv 1954	. . . . . . 7 ⊢ (𝜑 → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦)))
13	12	rexbidv 3191	. . . . . 6 ⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦)))
14	8	seqeq3d 14063	. . . . . . 7 ⊢ (𝜑 → seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))))
15	14	breq1d 5121	. . . . . 6 ⊢ (𝜑 → (seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑥))
16	2, 13, 15	3anbi123d 1464	. . . . 5 ⊢ (𝜑 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑥)))
17	16	rexbidv 3191	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑥)))
18	1	f1oeq3d 6821	. . . . . . 7 ⊢ (𝜑 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑚)–1-1-onto→𝐵))
19	6	cbvcsbdavw 36828	. . . . . . . . . . 11 ⊢ (𝜑 → ⦋(𝑓‘𝑛) / 𝑗⦌𝐶 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐷)
20	19	mpteq2dv 5207	. . . . . . . . . 10 ⊢ (𝜑 → (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))
21	20	seqeq3d 14063	. . . . . . . . 9 ⊢ (𝜑 → seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶)) = seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷)))
22	21	fveq1d 6887	. . . . . . . 8 ⊢ (𝜑 → (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))
23	22	eqeq2d 2776	. . . . . . 7 ⊢ (𝜑 → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚)))
24	18, 23	anbi12d 644	. . . . . 6 ⊢ (𝜑 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
25	24	exbidv 1954	. . . . 5 ⊢ (𝜑 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
26	25	rexbidv 3191	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
27	17, 26	orbi12d 932	. . 3 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚)))))
28	27	iotabidv 6524	. 2 ⊢ (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚)))))
29		df-prod 15981	. 2 ⊢ ∏𝑗 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑗⦌𝐶))‘𝑚))))
30		df-prod 15981	. 2 ⊢ ∏𝑘 ∈ 𝐵 𝐷 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐵, 𝐷, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐵 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐷))‘𝑚))))
31	28, 29, 30	3eqtr4g 2825	1 ⊢ (𝜑 → ∏𝑗 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091  ⦋csb 3854   ⊆ wss 3906  ifcif 4489   class class class wbr 5111   ↦ cmpt 5194  ℩cio 6494  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7419  0cc0 11115  1c1 11116   · cmul 11120  ℕcn 12248  ℤcz 12606  ℤ_≥cuz 12878  ...cfz 13551  seqcseq 14055   ⇝ cli 15559  ∏cprod 15980

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-xp 5669  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seq 14056  df-prod 15981

This theorem is used by: (None)