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Theorem cbvprod 14869
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 252 . . . . . 6 (𝐴 ⊆ (ℤ𝑚) ↔ 𝐴 ⊆ (ℤ𝑚))
2 cbvprod.2 . . . . . . . . . . . . . 14 𝑘𝐴
32nfcri 2949 . . . . . . . . . . . . 13 𝑘 𝑗𝐴
4 cbvprod.4 . . . . . . . . . . . . 13 𝑘𝐵
5 nfcv 2955 . . . . . . . . . . . . 13 𝑘1
63, 4, 5nfif 4315 . . . . . . . . . . . 12 𝑘if(𝑗𝐴, 𝐵, 1)
7 cbvprod.3 . . . . . . . . . . . . . 14 𝑗𝐴
87nfcri 2949 . . . . . . . . . . . . 13 𝑗 𝑘𝐴
9 cbvprod.5 . . . . . . . . . . . . 13 𝑗𝐶
10 nfcv 2955 . . . . . . . . . . . . 13 𝑗1
118, 9, 10nfif 4315 . . . . . . . . . . . 12 𝑗if(𝑘𝐴, 𝐶, 1)
12 eleq1w 2875 . . . . . . . . . . . . 13 (𝑗 = 𝑘 → (𝑗𝐴𝑘𝐴))
13 cbvprod.1 . . . . . . . . . . . . 13 (𝑗 = 𝑘𝐵 = 𝐶)
1412, 13ifbieq1d 4309 . . . . . . . . . . . 12 (𝑗 = 𝑘 → if(𝑗𝐴, 𝐵, 1) = if(𝑘𝐴, 𝐶, 1))
156, 11, 14cbvmpt 4950 . . . . . . . . . . 11 (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))
16 seqeq3 13032 . . . . . . . . . . 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 4858 . . . . . . . . 9 (seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)
1918anbi2i 611 . . . . . . . 8 ((𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
2019exbii 1933 . . . . . . 7 (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
2120rexbii 3236 . . . . . 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 13032 . . . . . . . 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 4858 . . . . . 6 (seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)
251, 21, 243anbi123i 1187 . . . . 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 3236 . . . 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 3740 . . . . . . . . . . 11 (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶
2827mpteq2i 4942 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)
29 seqeq3 13032 . . . . . . . . . 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 6412 . . . . . . . 8 (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)
3231eqeq2i 2825 . . . . . . 7 (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))
3332anbi2i 611 . . . . . 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 1933 . . . . 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 3236 . . . 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 929 . . 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 6089 . 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 14860 . 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 14860 . 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 2845 1 𝑗𝐴 𝐵 = ∏𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wo 865  w3a 1100   = wceq 1637  wex 1859  wcel 2157  wnfc 2942  wne 2985  wrex 3104  csb 3735  wss 3776  ifcif 4286   class class class wbr 4851  cmpt 4930  cio 6065  1-1-ontowf1o 6103  cfv 6104  (class class class)co 6877  0cc0 10224  1c1 10225   · cmul 10229  cn 11308  cz 11646  cuz 11907  ...cfz 12552  seqcseq 13027  cli 14441  cprod 14859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-br 4852  df-opab 4914  df-mpt 4931  df-xp 5324  df-cnv 5326  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-iota 6067  df-fv 6112  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-seq 13028  df-prod 14860
This theorem is referenced by:  cbvprodv  14870  cbvprodi  14871  fprodcllemf  14912  fproddivf  14941  fprodsplitf  14942  vonn0ioo2  41387  vonn0icc2  41389
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