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Theorem nfsum 11126
 Description: Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘 ∈ 𝐴𝐵. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jun-2019.)
Hypotheses
Ref Expression
nfsum.1 𝑥𝐴
nfsum.2 𝑥𝐵
Assertion
Ref Expression
nfsum 𝑥Σ𝑘𝐴 𝐵

Proof of Theorem nfsum
Dummy variables 𝑓 𝑗 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sumdc 11123 . 2 Σ𝑘𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
2 nfcv 2281 . . . . 5 𝑥
3 nfsum.1 . . . . . . 7 𝑥𝐴
4 nfcv 2281 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3090 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
63nfcri 2275 . . . . . . . 8 𝑥 𝑗𝐴
76nfdc 1637 . . . . . . 7 𝑥DECID 𝑗𝐴
84, 7nfralxy 2471 . . . . . 6 𝑥𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴
9 nfcv 2281 . . . . . . . 8 𝑥𝑚
10 nfcv 2281 . . . . . . . 8 𝑥 +
113nfcri 2275 . . . . . . . . . 10 𝑥 𝑛𝐴
12 nfcv 2281 . . . . . . . . . . 11 𝑥𝑛
13 nfsum.2 . . . . . . . . . . 11 𝑥𝐵
1412, 13nfcsb 3037 . . . . . . . . . 10 𝑥𝑛 / 𝑘𝐵
15 nfcv 2281 . . . . . . . . . 10 𝑥0
1611, 14, 15nfif 3500 . . . . . . . . 9 𝑥if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
172, 16nfmpt 4020 . . . . . . . 8 𝑥(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
189, 10, 17nfseq 10228 . . . . . . 7 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
19 nfcv 2281 . . . . . . 7 𝑥
20 nfcv 2281 . . . . . . 7 𝑥𝑧
2118, 19, 20nfbr 3974 . . . . . 6 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧
225, 8, 21nf3an 1545 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
232, 22nfrexxy 2472 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
24 nfcv 2281 . . . . 5 𝑥
25 nfcv 2281 . . . . . . . 8 𝑥𝑓
26 nfcv 2281 . . . . . . . 8 𝑥(1...𝑚)
2725, 26, 3nff1o 5365 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
28 nfcv 2281 . . . . . . . . . 10 𝑥1
29 nfv 1508 . . . . . . . . . . . 12 𝑥 𝑛𝑚
30 nfcv 2281 . . . . . . . . . . . . 13 𝑥(𝑓𝑛)
3130, 13nfcsb 3037 . . . . . . . . . . . 12 𝑥(𝑓𝑛) / 𝑘𝐵
3229, 31, 15nfif 3500 . . . . . . . . . . 11 𝑥if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)
3324, 32nfmpt 4020 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
3428, 10, 33nfseq 10228 . . . . . . . . 9 𝑥seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))
3534, 9nffv 5431 . . . . . . . 8 𝑥(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
3635nfeq2 2293 . . . . . . 7 𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)
3727, 36nfan 1544 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3837nfex 1616 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
3924, 38nfrexxy 2472 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))
4023, 39nfor 1553 . . 3 𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚)))
4140nfiotaw 5092 . 2 𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)))‘𝑚))))
421, 41nfcxfr 2278 1 𝑥Σ𝑘𝐴 𝐵
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ∨ wo 697  DECID wdc 819   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480  Ⅎwnfc 2268  ∀wral 2416  ∃wrex 2417  ⦋csb 3003   ⊆ wss 3071  ifcif 3474   class class class wbr 3929   ↦ cmpt 3989  ℩cio 5086  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  0cc0 7620  1c1 7621   + caddc 7623   ≤ cle 7801  ℕcn 8720  ℤcz 9054  ℤ≥cuz 9326  ...cfz 9790  seqcseq 10218   ⇝ cli 11047  Σcsu 11122 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 10219  df-sumdc 11123 This theorem is referenced by:  fsum2dlemstep  11203  fisumcom2  11207  fsumiun  11246  fsumcncntop  12725
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