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Theorem cbvsum 14273
 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, 3cbvcsb 3504 . . . . . . . . . . . 12 𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶
54a1i 11 . . . . . . . . . . 11 (⊤ → 𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶)
65ifeq1d 4054 . . . . . . . . . 10 (⊤ → if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
76mpteq2dv 4673 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
87seqeq3d 12671 . . . . . . . 8 (⊤ → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))))
98trud 1484 . . . . . . 7 seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
109breq1i 4590 . . . . . 6 (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)
1110anbi2i 726 . . . . 5 ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
1211rexbii 3023 . . . 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, 3cbvcsb 3504 . . . . . . . . . . . . 13 (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶
1413a1i 11 . . . . . . . . . . . 12 (⊤ → (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶)
1514mpteq2dv 4673 . . . . . . . . . . 11 (⊤ → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))
1615seqeq3d 12671 . . . . . . . . . 10 (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)))
1716trud 1484 . . . . . . . . 9 seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))
1817fveq1i 6104 . . . . . . . 8 (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)
1918eqeq2i 2622 . . . . . . 7 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))
2019anbi2i 726 . . . . . 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 1764 . . . . 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 3023 . . . 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 542 . . 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 5790 . 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 14265 . 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 14265 . 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 2642 1 Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1475  ⊤wtru 1476  ∃wex 1695   ∈ wcel 1977  Ⅎwnfc 2738  ∃wrex 2897  ⦋csb 3499   ⊆ wss 3540  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  ℩cio 5766  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕcn 10897  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  seqcseq 12663   ⇝ cli 14063  Σcsu 14264 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-sum 14265 This theorem is referenced by:  cbvsumv  14274  cbvsumi  14275  esumpfinvalf  29465  fsumclf  38633  fsumsplitf  38634  fsummulc1f  38635  fsumf1of  38641  fsumiunss  38642  fsumreclf  38643  fsumlessf  38644  fsumsermpt  38646  dvnmul  38833  sge0revalmpt  39271  sge0fsummpt  39283  sge0iunmptlemfi  39306  sge0iunmptlemre  39308  sge0ltfirpmpt2  39319  sge0isummpt2  39325  sge0xaddlem2  39327  sge0fsummptf  39329
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