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Theorem cbvproddavw2 36239
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 (𝜑𝐴 = 𝐵)
cbvproddavw2.2 ((𝜑𝑗 = 𝑘) → 𝐶 = 𝐷)
Assertion
Ref Expression
cbvproddavw2 (𝜑 → ∏𝑗𝐴 𝐶 = ∏𝑘𝐵 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.
StepHypRef Expression
1 cbvproddavw2.1 . . . . . . 7 (𝜑𝐴 = 𝐵)
21sseq1d 4027 . . . . . 6 (𝜑 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐵 ⊆ (ℤ𝑚)))
3 simpr 484 . . . . . . . . . . . . . 14 ((𝜑𝑗 = 𝑘) → 𝑗 = 𝑘)
41adantr 480 . . . . . . . . . . . . . 14 ((𝜑𝑗 = 𝑘) → 𝐴 = 𝐵)
53, 4eleq12d 2831 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑘) → (𝑗𝐴𝑘𝐵))
6 cbvproddavw2.2 . . . . . . . . . . . . 13 ((𝜑𝑗 = 𝑘) → 𝐶 = 𝐷)
75, 6ifbieq1d 4554 . . . . . . . . . . . 12 ((𝜑𝑗 = 𝑘) → if(𝑗𝐴, 𝐶, 1) = if(𝑘𝐵, 𝐷, 1))
87cbvmptdavw 36210 . . . . . . . . . . 11 (𝜑 → (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1)))
98seqeq3d 14036 . . . . . . . . . 10 (𝜑 → seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))))
109breq1d 5159 . . . . . . . . 9 (𝜑 → (seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦))
1110anbi2d 629 . . . . . . . 8 (𝜑 → ((𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦)))
1211exbidv 1917 . . . . . . 7 (𝜑 → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦)))
1312rexbidv 3175 . . . . . 6 (𝜑 → (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦)))
148seqeq3d 14036 . . . . . . 7 (𝜑 → seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))))
1514breq1d 5159 . . . . . 6 (𝜑 → (seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑥))
162, 13, 153anbi123d 1434 . . . . 5 (𝜑 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑥)))
1716rexbidv 3175 . . . 4 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑥)))
181f1oeq3d 6840 . . . . . . 7 (𝜑 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)–1-1-onto𝐵))
196cbvcsbdavw 36202 . . . . . . . . . . 11 (𝜑(𝑓𝑛) / 𝑗𝐶 = (𝑓𝑛) / 𝑘𝐷)
2019mpteq2dv 5251 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))
2120seqeq3d 14036 . . . . . . . . 9 (𝜑 → seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶)) = seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷)))
2221fveq1d 6903 . . . . . . . 8 (𝜑 → (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚))
2322eqeq2d 2744 . . . . . . 7 (𝜑 → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚)))
2418, 23anbi12d 631 . . . . . 6 (𝜑 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚))))
2524exbidv 1917 . . . . 5 (𝜑 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚))))
2625rexbidv 3175 . . . 4 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚))))
2717, 26orbi12d 917 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚)))))
2827iotabidv 6542 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚)))))
29 df-prod 15926 . 2 𝑗𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑗 ∈ ℤ ↦ if(𝑗𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))))
30 df-prod 15926 . 2 𝑘𝐵 𝐷 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐵, 𝐷, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚))))
3128, 29, 303eqtr4g 2798 1 (𝜑 → ∏𝑗𝐴 𝐶 = ∏𝑘𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846  w3a 1085   = wceq 1535  wex 1774  wcel 2104  wne 2936  wrex 3066  csb 3908  wss 3963  ifcif 4530   class class class wbr 5149  cmpt 5232  cio 6508  1-1-ontowf1o 6557  cfv 6558  (class class class)co 7425  0cc0 11146  1c1 11147   · cmul 11151  cn 12257  cz 12604  cuz 12869  ...cfz 13537  seqcseq 14028  cli 15506  cprod 15925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6317  df-iota 6510  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7428  df-oprab 7429  df-mpo 7430  df-frecs 8299  df-wrecs 8330  df-recs 8404  df-rdg 8443  df-seq 14029  df-prod 15926
This theorem is referenced by: (None)
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