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Theorem cbvsumdavw2 36238
Description: Change bound variable and the set of integers in a sum. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvsumdavw2.1 (𝜑𝐴 = 𝐵)
cbvsumdavw2.2 ((𝜑𝑗 = 𝑘) → 𝐶 = 𝐷)
Assertion
Ref Expression
cbvsumdavw2 (𝜑 → Σ𝑗𝐴 𝐶 = Σ𝑘𝐵 𝐷)
Distinct variable groups:   𝜑,𝑗,𝑘   𝐶,𝑘   𝐷,𝑗
Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗)   𝐷(𝑘)

Proof of Theorem cbvsumdavw2
Dummy variables 𝑥 𝑚 𝑛 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvsumdavw2.1 . . . . . . 7 (𝜑𝐴 = 𝐵)
21sseq1d 4027 . . . . . 6 (𝜑 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐵 ⊆ (ℤ𝑚)))
31eleq2d 2823 . . . . . . . . . 10 (𝜑 → (𝑛𝐴𝑛𝐵))
4 cbvsumdavw2.2 . . . . . . . . . . 11 ((𝜑𝑗 = 𝑘) → 𝐶 = 𝐷)
54cbvcsbdavw 36202 . . . . . . . . . 10 (𝜑𝑛 / 𝑗𝐶 = 𝑛 / 𝑘𝐷)
63, 5ifbieq1d 4554 . . . . . . . . 9 (𝜑 → if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0) = if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))
76mpteq2dv 5251 . . . . . . . 8 (𝜑 → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0)))
87seqeq3d 14036 . . . . . . 7 (𝜑 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))))
98breq1d 5159 . . . . . 6 (𝜑 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))) ⇝ 𝑥))
102, 9anbi12d 631 . . . . 5 (𝜑 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))) ⇝ 𝑥)))
1110rexbidv 3175 . . . 4 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))) ⇝ 𝑥)))
121f1oeq3d 6840 . . . . . . 7 (𝜑 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)–1-1-onto𝐵))
134cbvcsbdavw 36202 . . . . . . . . . . 11 (𝜑(𝑓𝑛) / 𝑗𝐶 = (𝑓𝑛) / 𝑘𝐷)
1413mpteq2dv 5251 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))
1514seqeq3d 14036 . . . . . . . . 9 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷)))
1615fveq1d 6903 . . . . . . . 8 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚))
1716eqeq2d 2744 . . . . . . 7 (𝜑 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚)))
1812, 17anbi12d 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))))
1918exbidv 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))))
2019rexbidv 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))))
2111, 20orbi12d 917 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚)))))
2221iotabidv 6542 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚)))))
23 df-sum 15709 . 2 Σ𝑗𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))))
24 df-sum 15709 . 2 Σ𝑘𝐵 𝐷 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐷, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐷))‘𝑚))))
2522, 23, 243eqtr4g 2798 1 (𝜑 → Σ𝑗𝐴 𝐶 = Σ𝑘𝐵 𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 846   = wceq 1535  wex 1774  wcel 2104  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   + caddc 11149  cn 12257  cz 12604  cuz 12869  ...cfz 13537  seqcseq 14028  cli 15506  Σcsu 15708
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-sum 15709
This theorem is referenced by: (None)
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