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Theorem cbvsum 11239
 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

Proof of Theorem cbvsum
Dummy variables 𝑓 𝑚 𝑛 𝑢 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cbvsum.4 . . . . . . . . . . 11 𝑘𝐵
2 cbvsum.5 . . . . . . . . . . 11 𝑗𝐶
3 cbvsum.1 . . . . . . . . . . 11 (𝑗 = 𝑘𝐵 = 𝐶)
41, 2, 3cbvcsb 3036 . . . . . . . . . 10 𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶
5 ifeq1 3508 . . . . . . . . . 10 (𝑛 / 𝑗𝐵 = 𝑛 / 𝑘𝐶 → if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
64, 5ax-mp 5 . . . . . . . . 9 if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)
76mpteq2i 4051 . . . . . . . 8 (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
8 seqeq3 10331 . . . . . . . 8 ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))))
97, 8ax-mp 5 . . . . . . 7 seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
109breq1i 3972 . . . . . 6 (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)
11103anbi3i 1175 . . . . 5 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
1211rexbii 2464 . . . 4 (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
131, 2, 3cbvcsb 3036 . . . . . . . . . . . 12 (𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶
14 ifeq1 3508 . . . . . . . . . . . 12 ((𝑓𝑛) / 𝑗𝐵 = (𝑓𝑛) / 𝑘𝐶 → if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0))
1513, 14ax-mp 5 . . . . . . . . . . 11 if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)
1615mpteq2i 4051 . . . . . . . . . 10 (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0))
17 seqeq3 10331 . . . . . . . . . 10 ((𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0))))
1816, 17ax-mp 5 . . . . . . . . 9 seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))
1918fveq1i 5466 . . . . . . . 8 (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)
2019eqeq2i 2168 . . . . . . 7 (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))
2120anbi2i 453 . . . . . 6 ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)))
2221exbii 1585 . . . . 5 (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)))
2322rexbii 2464 . . . 4 (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)))
2412, 23orbi12i 754 . . 3 ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))))
2524iotabii 5154 . 2 (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))))
26 df-sumdc 11233 . 2 Σ𝑗𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑗𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑗𝐵, 0)))‘𝑚))))
27 df-sumdc 11233 . 2 Σ𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑢 ∈ (ℤ𝑚)DECID 𝑢𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))))
2825, 26, 273eqtr4i 2188 1 Σ𝑗𝐴 𝐵 = Σ𝑘𝐴 𝐶
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698  DECID wdc 820   ∧ w3a 963   = wceq 1335  ∃wex 1472   ∈ wcel 2128  Ⅎwnfc 2286  ∀wral 2435  ∃wrex 2436  ⦋csb 3031   ⊆ wss 3102  ifcif 3505   class class class wbr 3965   ↦ cmpt 4025  ℩cio 5130  –1-1-onto→wf1o 5166  ‘cfv 5167  (class class class)co 5818  0cc0 7715  1c1 7716   + caddc 7718   ≤ cle 7896  ℕcn 8816  ℤcz 9150  ℤ≥cuz 9422  ...cfz 9894  seqcseq 10326   ⇝ cli 11157  Σcsu 11232 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-if 3506  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-cnv 4591  df-dm 4593  df-rn 4594  df-res 4595  df-iota 5132  df-fv 5175  df-ov 5821  df-oprab 5822  df-mpo 5823  df-recs 6246  df-frec 6332  df-seqfrec 10327  df-sumdc 11233 This theorem is referenced by:  cbvsumv  11240  cbvsumi  11241  fsumsplitf  11287
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