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Theorem nfsum 10710
Description: Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘𝐴𝐵. (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-isum 10707 . 2 Σ𝑘𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
2 nfcv 2228 . . . . 5 𝑥
3 nfsum.1 . . . . . . 7 𝑥𝐴
4 nfcv 2228 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3016 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
63nfcri 2222 . . . . . . . 8 𝑥 𝑗𝐴
76nfdc 1594 . . . . . . 7 𝑥DECID 𝑗𝐴
84, 7nfralxy 2414 . . . . . 6 𝑥𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴
9 nfcv 2228 . . . . . . . 8 𝑥𝑚
10 nfcv 2228 . . . . . . . 8 𝑥 +
113nfcri 2222 . . . . . . . . . 10 𝑥 𝑛𝐴
12 nfcv 2228 . . . . . . . . . . 11 𝑥𝑛
13 nfsum.2 . . . . . . . . . . 11 𝑥𝐵
1412, 13nfcsb 2963 . . . . . . . . . 10 𝑥𝑛 / 𝑘𝐵
15 nfcv 2228 . . . . . . . . . 10 𝑥0
1611, 14, 15nfif 3415 . . . . . . . . 9 𝑥if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
172, 16nfmpt 3922 . . . . . . . 8 𝑥(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
18 nfcv 2228 . . . . . . . 8 𝑥
199, 10, 17, 18nfiseq 9833 . . . . . . 7 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ)
20 nfcv 2228 . . . . . . 7 𝑥
21 nfcv 2228 . . . . . . 7 𝑥𝑧
2219, 20, 21nfbr 3881 . . . . . 6 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧
235, 8, 22nf3an 1503 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧)
242, 23nfrexxy 2415 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧)
25 nfcv 2228 . . . . 5 𝑥
26 nfcv 2228 . . . . . . . 8 𝑥𝑓
27 nfcv 2228 . . . . . . . 8 𝑥(1...𝑚)
2826, 27, 3nff1o 5235 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
29 nfcv 2228 . . . . . . . . . 10 𝑥1
30 nfv 1466 . . . . . . . . . . . 12 𝑥 𝑛𝑚
31 nfcv 2228 . . . . . . . . . . . . 13 𝑥(𝑓𝑛)
3231, 13nfcsb 2963 . . . . . . . . . . . 12 𝑥(𝑓𝑛) / 𝑘𝐵
3330, 32, 15nfif 3415 . . . . . . . . . . 11 𝑥if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)
3425, 33nfmpt 3922 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0))
3529, 10, 34, 18nfiseq 9833 . . . . . . . . 9 𝑥seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)
3635, 9nffv 5299 . . . . . . . 8 𝑥(seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)
3736nfeq2 2240 . . . . . . 7 𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)
3828, 37nfan 1502 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
3938nfex 1573 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
4025, 39nfrexxy 2415 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))
4124, 40nfor 1511 . . 3 𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))
4241nfiotaxy 4971 . 2 𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
431, 42nfcxfr 2225 1 𝑥Σ𝑘𝐴 𝐵
Colors of variables: wff set class
Syntax hints:  wa 102  wo 664  DECID wdc 780  w3a 924   = wceq 1289  wex 1426  wcel 1438  wnfc 2215  wral 2359  wrex 2360  csb 2931  wss 2997  ifcif 3389   class class class wbr 3837  cmpt 3891  cio 4965  1-1-ontowf1o 5001  cfv 5002  (class class class)co 5634  cc 7327  0cc0 7329  1c1 7330   + caddc 7332  cle 7502  cn 8394  cz 8720  cuz 8988  ...cfz 9393  seqcseq4 9816  cli 10630  Σcsu 10706
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-un 3001  df-in 3003  df-ss 3010  df-if 3390  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-mpt 3893  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-recs 6052  df-frec 6138  df-iseq 9818  df-isum 10707
This theorem is referenced by:  fsum2dlemstep  10791  fisumcom2  10795  fsumiun  10833
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