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Theorem nfsum1 11137
Description: Bound-variable hypothesis builder for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypothesis
Ref Expression
nfsum1.1 𝑘𝐴
Assertion
Ref Expression
nfsum1 𝑘Σ𝑘𝐴 𝐵

Proof of Theorem nfsum1
Dummy variables 𝑓 𝑗 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sumdc 11135 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
2 nfcv 2281 . . . . 5 𝑘
3 nfsum1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2281 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 3090 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
63nfcri 2275 . . . . . . . 8 𝑘 𝑗𝐴
76nfdc 1637 . . . . . . 7 𝑘DECID 𝑗𝐴
84, 7nfralxy 2471 . . . . . 6 𝑘𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴
9 nfcv 2281 . . . . . . . 8 𝑘𝑚
10 nfcv 2281 . . . . . . . 8 𝑘 +
113nfcri 2275 . . . . . . . . . 10 𝑘 𝑛𝐴
12 nfcsb1v 3035 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
13 nfcv 2281 . . . . . . . . . 10 𝑘0
1411, 12, 13nfif 3500 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
152, 14nfmpt 4020 . . . . . . . 8 𝑘(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
169, 10, 15nfseq 10240 . . . . . . 7 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
17 nfcv 2281 . . . . . . 7 𝑘
18 nfcv 2281 . . . . . . 7 𝑘𝑥
1916, 17, 18nfbr 3974 . . . . . 6 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
205, 8, 19nf3an 1545 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
212, 20nfrexya 2474 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
22 nfcv 2281 . . . . 5 𝑘
23 nfcv 2281 . . . . . . . 8 𝑘𝑓
24 nfcv 2281 . . . . . . . 8 𝑘(1...𝑚)
2523, 24, 3nff1o 5365 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
26 nfcv 2281 . . . . . . . . . 10 𝑘1
27 nfv 1508 . . . . . . . . . . . 12 𝑘 𝑛𝑚
28 nfcsb1v 3035 . . . . . . . . . . . 12 𝑘(𝑓𝑛) / 𝑘𝐵
2927, 28, 13nfif 3500 . . . . . . . . . . 11 𝑘if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)
3022, 29nfmpt 4020 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
3126, 10, 30nfseq 10240 . . . . . . . . 9 𝑘seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))
3231, 9nffv 5431 . . . . . . . 8 𝑘(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
3332nfeq2 2293 . . . . . . 7 𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
3425, 33nfan 1544 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3534nfex 1616 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3622, 35nfrexya 2474 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3721, 36nfor 1553 . . 3 𝑘(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
3837nfiotaw 5092 . 2 𝑘(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
391, 38nfcxfr 2278 1 𝑘Σ𝑘𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wa 103  wo 697  DECID wdc 819  w3a 962   = wceq 1331  wex 1468  wcel 1480  wnfc 2268  wral 2416  wrex 2417  csb 3003  wss 3071  ifcif 3474   class class class wbr 3929  cmpt 3989  cio 5086  1-1-ontowf1o 5122  cfv 5123  (class class class)co 5774  0cc0 7632  1c1 7633   + caddc 7635  cle 7813  cn 8732  cz 9066  cuz 9338  ...cfz 9802  seqcseq 10230  cli 11059  Σcsu 11134
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-seqfrec 10231  df-sumdc 11135
This theorem is referenced by:  mertenslem2  11317
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