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Theorem cbvsumdavw 36258
Description: Change bound variable in a sum. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypothesis
Ref Expression
cbvsumdavw.1 ((𝜑𝑘 = 𝑗) → 𝐵 = 𝐶)
Assertion
Ref Expression
cbvsumdavw (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑗𝐴 𝐶)
Distinct variable groups:   𝜑,𝑘,𝑗   𝐵,𝑗   𝐶,𝑘
Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑘)   𝐶(𝑗)

Proof of Theorem cbvsumdavw
Dummy variables 𝑥 𝑚 𝑛 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvsumdavw.1 . . . . . . . . . . 11 ((𝜑𝑘 = 𝑗) → 𝐵 = 𝐶)
21cbvcsbdavw 36238 . . . . . . . . . 10 (𝜑𝑛 / 𝑘𝐵 = 𝑛 / 𝑗𝐶)
32ifeq1d 4543 . . . . . . . . 9 (𝜑 → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))
43mpteq2dv 5242 . . . . . . . 8 (𝜑 → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0)))
54seqeq3d 14046 . . . . . . 7 (𝜑 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))))
65breq1d 5151 . . . . . 6 (𝜑 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥))
76anbi2d 630 . . . . 5 (𝜑 → ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥)))
87rexbidv 3178 . . . 4 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐶, 0))) ⇝ 𝑥)))
91cbvcsbdavw 36238 . . . . . . . . . . 11 (𝜑(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑗𝐶)
109mpteq2dv 5242 . . . . . . . . . 10 (𝜑 → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))
1110seqeq3d 14046 . . . . . . . . 9 (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶)))
1211fveq1d 6906 . . . . . . . 8 (𝜑 → (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))
1312eqeq2d 2747 . . . . . . 7 (𝜑 → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚)))
1413anbi2d 630 . . . . . 6 (𝜑 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))))
1514exbidv 1921 . . . . 5 (𝜑 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))))
1615rexbidv 3178 . . . 4 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐶))‘𝑚))))
178, 16orbi12d 919 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ↔ (∃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)))))
1817iotabidv 6543 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))) = (℩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)))))
19 df-sum 15719 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
20 df-sum 15719 . 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))))
2118, 19, 203eqtr4g 2801 1 (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑗𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wex 1779  wcel 2108  wrex 3069  csb 3898  wss 3950  ifcif 4524   class class class wbr 5141  cmpt 5223  cio 6510  1-1-ontowf1o 6558  cfv 6559  (class class class)co 7429  0cc0 11151  1c1 11152   + caddc 11154  cn 12262  cz 12609  cuz 12874  ...cfz 13543  seqcseq 14038  cli 15516  Σcsu 15718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-mpt 5224  df-xp 5689  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-pred 6319  df-iota 6512  df-fv 6567  df-ov 7432  df-oprab 7433  df-mpo 7434  df-frecs 8302  df-wrecs 8333  df-recs 8407  df-rdg 8446  df-seq 14039  df-sum 15719
This theorem is referenced by: (None)
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