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Theorem cbvsum 15259
Description: Change bound variable in a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypotheses
Ref Expression
cbvsum.1 (𝑗 = 𝑘𝐵 = 𝐶)
cbvsum.2 𝑘𝐴
cbvsum.3 𝑗𝐴
cbvsum.4 𝑘𝐵
cbvsum.5 𝑗𝐶
Assertion
Ref Expression
cbvsum Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
Distinct variable group:   𝑗,𝑘
Allowed substitution hints:   𝐴(𝑗,𝑘)   𝐵(𝑗,𝑘)   𝐶(𝑗,𝑘)

Proof of Theorem cbvsum
Dummy variables 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvsum.4 . . . . . . . . . . . . 13 𝑘𝐵
2 cbvsum.5 . . . . . . . . . . . . 13 𝑗𝐶
3 cbvsum.1 . . . . . . . . . . . . 13 (𝑗 = 𝑘𝐵 = 𝐶)
41, 2, 3cbvcsbw 3821 . . . . . . . . . . . 12 𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶
54a1i 11 . . . . . . . . . . 11 (⊤ → 𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶)
65ifeq1d 4458 . . . . . . . . . 10 (⊤ → if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
76mpteq2dv 5151 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
87seqeq3d 13582 . . . . . . . 8 (⊤ → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))))
98mptru 1550 . . . . . . 7 seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
109breq1i 5060 . . . . . 6 (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)
1110anbi2i 626 . . . . 5 ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
1211rexbii 3170 . . . 4 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
131, 2, 3cbvcsbw 3821 . . . . . . . . . . . . 13 (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶
1413a1i 11 . . . . . . . . . . . 12 (⊤ → (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶)
1514mpteq2dv 5151 . . . . . . . . . . 11 (⊤ → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))
1615seqeq3d 13582 . . . . . . . . . 10 (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)))
1716mptru 1550 . . . . . . . . 9 seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))
1817fveq1i 6718 . . . . . . . 8 (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)
1918eqeq2i 2750 . . . . . . 7 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))
2019anbi2i 626 . . . . . 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)))
2120exbii 1855 . . . . 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)))
2221rexbii 3170 . . . 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)))
2312, 22orbi12i 915 . . 3 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
2423iotabii 6365 . 2 (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
25 df-sum 15250 . 2 Σ𝑗𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚))))
26 df-sum 15250 . 2 Σ𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
2724, 25, 263eqtr4i 2775 1 Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wo 847   = wceq 1543  wtru 1544  wex 1787  wcel 2110  wnfc 2884  wrex 3062  csb 3811  wss 3866  ifcif 4439   class class class wbr 5053  cmpt 5135  cio 6336  1-1-ontowf1o 6379  cfv 6380  (class class class)co 7213  0cc0 10729  1c1 10730   + caddc 10732  cn 11830  cz 12176  cuz 12438  ...cfz 13095  seqcseq 13574  cli 15045  Σcsu 15249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-xp 5557  df-cnv 5559  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-iota 6338  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-seq 13575  df-sum 15250
This theorem is referenced by:  cbvsumv  15260  cbvsumi  15261  fsumclf  15302  fsumiunle  30863  esumpfinvalf  31756  sticksstones8  39831  sticksstones10  39833  sticksstones12a  39835  sticksstones12  39836  fsummulc1f  42787  fsumf1of  42790  fsumiunss  42791  fsumreclf  42792  fsumlessf  42793  fsumsermpt  42795  dvnmul  43159  sge0revalmpt  43591  sge0fsummpt  43603  sge0iunmptlemfi  43626  sge0iunmptlemre  43628  sge0ltfirpmpt2  43639  sge0isummpt2  43645  sge0xaddlem2  43647  sge0fsummptf  43649
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