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Theorem cbvsum 15064
 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 3840 . . . . . . . . . . . 12 𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶
54a1i 11 . . . . . . . . . . 11 (⊤ → 𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶)
65ifeq1d 4446 . . . . . . . . . 10 (⊤ → if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
76mpteq2dv 5130 . . . . . . . . 9 (⊤ → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
87seqeq3d 13392 . . . . . . . 8 (⊤ → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))))
98mptru 1545 . . . . . . 7 seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
109breq1i 5041 . . . . . 6 (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)
1110anbi2i 625 . . . . 5 ((𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
1211rexbii 3211 . . . 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 3840 . . . . . . . . . . . . 13 (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶
1413a1i 11 . . . . . . . . . . . 12 (⊤ → (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶)
1514mpteq2dv 5130 . . . . . . . . . . 11 (⊤ → (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))
1615seqeq3d 13392 . . . . . . . . . 10 (⊤ → seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)))
1716mptru 1545 . . . . . . . . 9 seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵)) = seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))
1817fveq1i 6656 . . . . . . . 8 (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)
1918eqeq2i 2811 . . . . . . 7 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑗𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))
2019anbi2i 625 . . . . . 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 1849 . . . . 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 3211 . . . 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 912 . . 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 6317 . 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 15055 . 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 15055 . 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 2831 1 Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538  ⊤wtru 1539  ∃wex 1781   ∈ wcel 2111  Ⅎwnfc 2936  ∃wrex 3107  ⦋csb 3830   ⊆ wss 3883  ifcif 4428   class class class wbr 5034   ↦ cmpt 5114  ℩cio 6289  –1-1-onto→wf1o 6331  ‘cfv 6332  (class class class)co 7145  0cc0 10544  1c1 10545   + caddc 10547  ℕcn 11643  ℤcz 11989  ℤ≥cuz 12251  ...cfz 12905  seqcseq 13384   ⇝ cli 14853  Σcsu 15054 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-un 3888  df-in 3890  df-ss 3900  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-xp 5529  df-cnv 5531  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-iota 6291  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-seq 13385  df-sum 15055 This theorem is referenced by:  cbvsumv  15065  cbvsumi  15066  fsumiunle  30615  esumpfinvalf  31511  fsumclf  42379  fsummulc1f  42380  fsumf1of  42384  fsumiunss  42385  fsumreclf  42386  fsumlessf  42387  fsumsermpt  42389  dvnmul  42753  sge0revalmpt  43185  sge0fsummpt  43197  sge0iunmptlemfi  43220  sge0iunmptlemre  43222  sge0ltfirpmpt2  43233  sge0isummpt2  43239  sge0xaddlem2  43241  sge0fsummptf  43243
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