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Theorem cbvprod 14484
 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 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 biid 250 . . . . . 6 (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑚))
2 cbvprod.2 . . . . . . . . . . . . . 14 𝑘𝐴
32nfcri 2745 . . . . . . . . . . . . 13 𝑘 𝑗𝐴
4 cbvprod.4 . . . . . . . . . . . . 13 𝑘𝐵
5 nfcv 2751 . . . . . . . . . . . . 13 𝑘1
63, 4, 5nfif 4065 . . . . . . . . . . . 12 𝑘if(𝑗𝐴, 𝐵, 1)
7 cbvprod.3 . . . . . . . . . . . . . 14 𝑗𝐴
87nfcri 2745 . . . . . . . . . . . . 13 𝑗 𝑘𝐴
9 cbvprod.5 . . . . . . . . . . . . 13 𝑗𝐶
10 nfcv 2751 . . . . . . . . . . . . 13 𝑗1
118, 9, 10nfif 4065 . . . . . . . . . . . 12 𝑗if(𝑘𝐴, 𝐶, 1)
12 eleq1 2676 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
13 cbvprod.1 . . . . . . . . . . . . 13 (𝑗 = 𝑘𝐵 = 𝐶)
1412, 13ifbieq1d 4059 . . . . . . . . . . . 12 (𝑗 = 𝑘 → if(𝑗𝐴, 𝐵, 1) = if(𝑘𝐴, 𝐶, 1))
156, 11, 14cbvmpt 4677 . . . . . . . . . . 11 (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))
16 seqeq3 12668 . . . . . . . . . . 11 ((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))))
1715, 16ax-mp 5 . . . . . . . . . 10 seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)))
1817breq1i 4590 . . . . . . . . 9 (seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)
1918anbi2i 726 . . . . . . . 8 ((𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
2019exbii 1764 . . . . . . 7 (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
2120rexbii 3023 . . . . . 6 (∃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))
22 seqeq3 12668 . . . . . . . 8 ((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))))
2315, 22ax-mp 5 . . . . . . 7 seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)))
2423breq1i 4590 . . . . . 6 (seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)
251, 21, 243anbi123i 1244 . . . . 5 ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥))
2625rexbii 3023 . . . 4 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥))
274, 9, 13cbvcsb 3504 . . . . . . . . . . 11 (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶
2827mpteq2i 4669 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)
29 seqeq3 12668 . . . . . . . . . 10 ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶) → seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)))
3028, 29ax-mp 5 . . . . . . . . 9 seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))
3130fveq1i 6104 . . . . . . . 8 (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)
3231eqeq2i 2622 . . . . . . 7 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))
3332anbi2i 726 . . . . . 6 ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))
3433exbii 1764 . . . . 5 (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))
3534rexbii 3023 . . . 4 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))
3626, 35orbi12i 542 . . 3 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
3736iotabii 5790 . 2 (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
38 df-prod 14475 . 2 𝑗𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚))))
39 df-prod 14475 . 2 𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃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 ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
4037, 38, 393eqtr4i 2642 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738   ≠ wne 2780  ∃wrex 2897  ⦋csb 3499   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ℩cio 5766  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   · cmul 9820  ℕcn 10897  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663   ⇝ cli 14063  ∏cprod 14474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-prod 14475 This theorem is referenced by:  cbvprodv  14485  cbvprodi  14486  fprodcllemf  14527  fproddivf  14557  fprodsplitf  14558  vonn0ioo2  39581  vonn0icc2  39583
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