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Theorem sumeq1 11377
Description: Equality theorem for a sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Assertion
Ref Expression
sumeq1 (𝐴 = 𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)

Proof of Theorem sumeq1
Dummy variables 𝑓 𝑚 𝑛 𝑥 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sseq1 3190 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐵 ⊆ (ℤ𝑚)))
2 eleq2 2251 . . . . . . . 8 (𝐴 = 𝐵 → (𝑗𝐴𝑗𝐵))
32dcbid 839 . . . . . . 7 (𝐴 = 𝐵 → (DECID 𝑗𝐴DECID 𝑗𝐵))
43ralbidv 2487 . . . . . 6 (𝐴 = 𝐵 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵))
5 simpl 109 . . . . . . . . . . 11 ((𝐴 = 𝐵𝑛 ∈ ℤ) → 𝐴 = 𝐵)
65eleq2d 2257 . . . . . . . . . 10 ((𝐴 = 𝐵𝑛 ∈ ℤ) → (𝑛𝐴𝑛𝐵))
76ifbid 3567 . . . . . . . . 9 ((𝐴 = 𝐵𝑛 ∈ ℤ) → if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0) = if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))
87mpteq2dva 4105 . . . . . . . 8 (𝐴 = 𝐵 → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)))
98seqeq3d 10467 . . . . . . 7 (𝐴 = 𝐵 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))))
109breq1d 4025 . . . . . 6 (𝐴 = 𝐵 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥))
111, 4, 103anbi123d 1322 . . . . 5 (𝐴 = 𝐵 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)))
1211rexbidv 2488 . . . 4 (𝐴 = 𝐵 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥)))
13 f1oeq3 5463 . . . . . . 7 (𝐴 = 𝐵 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)–1-1-onto𝐵))
1413anbi1d 465 . . . . . 6 (𝐴 = 𝐵 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))))
1514exbidv 1835 . . . . 5 (𝐴 = 𝐵 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))))
1615rexbidv 2488 . . . 4 (𝐴 = 𝐵 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))))
1712, 16orbi12d 794 . . 3 (𝐴 = 𝐵 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ 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))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)))))
1817iotabidv 5211 . 2 (𝐴 = 𝐵 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ 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)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚)))))
19 df-sumdc 11376 . 2 Σ𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ 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))))
20 df-sumdc 11376 . 2 Σ𝑘𝐵 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))‘𝑚))))
2118, 19, 203eqtr4g 2245 1 (𝐴 = 𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709  DECID wdc 835  w3a 979   = wceq 1363  wex 1502  wcel 2158  wral 2465  wrex 2466  csb 3069  wss 3141  ifcif 3546   class class class wbr 4015  cmpt 4076  cio 5188  1-1-ontowf1o 5227  cfv 5228  (class class class)co 5888  0cc0 7825  1c1 7826   + caddc 7828  cle 8007  cn 8933  cz 9267  cuz 9542  ...cfz 10022  seqcseq 10459  cli 11300  Σcsu 11375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-recs 6320  df-frec 6406  df-seqfrec 10460  df-sumdc 11376
This theorem is referenced by:  sumeq1i  11385  sumeq1d  11388  isumz  11411  fsumadd  11428  fsum2d  11457  fisumrev2  11468  fsummulc2  11470  fsumconst  11476  modfsummod  11480  fsumabs  11487  fsumrelem  11493  fsumiun  11499  fsumcncntop  14352
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