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Theorem sumeq1 10330
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 3021 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ⊆ (ℤ𝑚) ↔ 𝐵 ⊆ (ℤ𝑚)))
2 eleq2 2143 . . . . . . . 8 (𝐴 = 𝐵 → (𝑗𝐴𝑗𝐵))
32dcbid 782 . . . . . . 7 (𝐴 = 𝐵 → (DECID 𝑗𝐴DECID 𝑗𝐵))
43ralbidv 2369 . . . . . 6 (𝐴 = 𝐵 → (∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ↔ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵))
5 simpl 107 . . . . . . . . . . 11 ((𝐴 = 𝐵𝑛 ∈ ℤ) → 𝐴 = 𝐵)
65eleq2d 2149 . . . . . . . . . 10 ((𝐴 = 𝐵𝑛 ∈ ℤ) → (𝑛𝐴𝑛𝐵))
76ifbid 3378 . . . . . . . . 9 ((𝐴 = 𝐵𝑛 ∈ ℤ) → if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0) = if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0))
87mpteq2dva 3876 . . . . . . . 8 (𝐴 = 𝐵 → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)))
9 iseqeq3 9526 . . . . . . . 8 ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)) → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ))
108, 9syl 14 . . . . . . 7 (𝐴 = 𝐵 → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ))
1110breq1d 3803 . . . . . 6 (𝐴 = 𝐵 → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥))
121, 4, 113anbi123d 1244 . . . . 5 (𝐴 = 𝐵 → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ↔ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥)))
1312rexbidv 2370 . . . 4 (𝐴 = 𝐵 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥)))
14 f1oeq3 5150 . . . . . . 7 (𝐴 = 𝐵 → (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)–1-1-onto𝐵))
1514anbi1d 453 . . . . . 6 (𝐴 = 𝐵 → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
1615exbidv 1747 . . . . 5 (𝐴 = 𝐵 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
1716rexbidv 2370 . . . 4 (𝐴 = 𝐵 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
1813, 17orbi12d 740 . . 3 (𝐴 = 𝐵 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)))))
1918iotabidv 4918 . 2 (𝐴 = 𝐵 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)))))
20 df-isum 10329 . 2 Σ𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
21 df-isum 10329 . 2 Σ𝑘𝐵 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐵 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐵 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐵, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐵𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
2219, 20, 213eqtr4g 2139 1 (𝐴 = 𝐵 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662  DECID wdc 776  w3a 920   = wceq 1285  wex 1422  wcel 1434  wral 2349  wrex 2350  csb 2909  wss 2974  ifcif 3359   class class class wbr 3793  cmpt 3847  cio 4895  1-1-ontowf1o 4931  cfv 4932  (class class class)co 5543  cc 7041  0cc0 7043  1c1 7044   + caddc 7046  cle 7216  cn 8106  cz 8432  cuz 8700  ...cfz 9105  seqcseq 9521  cli 10255  Σcsu 10328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-if 3360  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-mpt 3849  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-iota 4897  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-recs 5954  df-frec 6040  df-iseq 9522  df-isum 10329
This theorem is referenced by: (None)
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