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Theorem cbvprod 11339
 Description: Change bound variable in a product. (Contributed by Scott Fenton, 4-Dec-2017.)
Hypotheses
Ref Expression
cbvprod.1 (𝑗 = 𝑘𝐵 = 𝐶)
cbvprod.2 𝑘𝐴
cbvprod.3 𝑗𝐴
cbvprod.4 𝑘𝐵
cbvprod.5 𝑗𝐶
Assertion
Ref Expression
cbvprod 𝑗𝐴 𝐵 = ∏𝑘𝐴 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 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvprod.2 . . . . . . . . . . . . . . 15 𝑘𝐴
21nfcri 2275 . . . . . . . . . . . . . 14 𝑘 𝑗𝐴
3 cbvprod.4 . . . . . . . . . . . . . 14 𝑘𝐵
4 nfcv 2281 . . . . . . . . . . . . . 14 𝑘1
52, 3, 4nfif 3500 . . . . . . . . . . . . 13 𝑘if(𝑗𝐴, 𝐵, 1)
6 cbvprod.3 . . . . . . . . . . . . . . 15 𝑗𝐴
76nfcri 2275 . . . . . . . . . . . . . 14 𝑗 𝑘𝐴
8 cbvprod.5 . . . . . . . . . . . . . 14 𝑗𝐶
9 nfcv 2281 . . . . . . . . . . . . . 14 𝑗1
107, 8, 9nfif 3500 . . . . . . . . . . . . 13 𝑗if(𝑘𝐴, 𝐶, 1)
11 eleq1w 2200 . . . . . . . . . . . . . 14 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
12 cbvprod.1 . . . . . . . . . . . . . 14 (𝑗 = 𝑘𝐵 = 𝐶)
1311, 12ifbieq1d 3494 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → if(𝑗𝐴, 𝐵, 1) = if(𝑘𝐴, 𝐶, 1))
145, 10, 13cbvmpt 4023 . . . . . . . . . . . 12 (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))
15 seqeq3 10235 . . . . . . . . . . . 12 ((𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)) → seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
1614, 15ax-mp 5 . . . . . . . . . . 11 seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)))
1716breq1i 3936 . . . . . . . . . 10 (seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)
1817anbi2i 452 . . . . . . . . 9 ((𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
1918exbii 1584 . . . . . . . 8 (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
2019rexbii 2442 . . . . . . 7 (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
21 seqeq3 10235 . . . . . . . . 9 ((𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)) → seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
2214, 21ax-mp 5 . . . . . . . 8 seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)))
2322breq1i 3936 . . . . . . 7 (seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)
2420, 23anbi12i 455 . . . . . 6 ((∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥))
2524anbi2i 452 . . . . 5 (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)))
2625rexbii 2442 . . . 4 (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)))
273, 8, 12cbvcsbw 3007 . . . . . . . . . . . 12 (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶
28 ifeq1 3477 . . . . . . . . . . . 12 ((𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶 → if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1))
2927, 28ax-mp 5 . . . . . . . . . . 11 if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)
3029mpteq2i 4015 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1))
31 seqeq3 10235 . . . . . . . . . 10 ((𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1))))
3230, 31ax-mp 5 . . . . . . . . 9 seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))
3332fveq1i 5422 . . . . . . . 8 (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚)
3433eqeq2i 2150 . . . . . . 7 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))
3534anbi2i 452 . . . . . 6 ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚)))
3635exbii 1584 . . . . 5 (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚)))
3736rexbii 2442 . . . 4 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚)))
3826, 37orbi12i 753 . . 3 ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))))
3938iotabii 5110 . 2 (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))))
40 df-proddc 11332 . 2 𝑗𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 1)))‘𝑚))))
41 df-proddc 11332 . 2 𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑠 ∈ (ℤ𝑚)DECID 𝑠𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))))
4239, 40, 413eqtr4i 2170 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 697  DECID wdc 819   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Ⅎwnfc 2268  ∀wral 2416  ∃wrex 2417  ⦋csb 3003   ⊆ wss 3071  ifcif 3474   class class class wbr 3929   ↦ cmpt 3989  ℩cio 5086  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  0cc0 7632  1c1 7633   · cmul 7637   ≤ cle 7813   # cap 8355  ℕcn 8732  ℤcz 9066  ℤ≥cuz 9338  ...cfz 9802  seqcseq 10230   ⇝ cli 11059  ∏cprod 11331 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-cnv 4547  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10231  df-proddc 11332 This theorem is referenced by:  cbvprodv  11340  cbvprodi  11341
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