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Theorem nfsum 14355
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 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 14351 . 2 Σ𝑘𝐴 𝐵 = (℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
2 nfcv 2761 . . . . 5 𝑥
3 nfsum.1 . . . . . . 7 𝑥𝐴
4 nfcv 2761 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3576 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2761 . . . . . . . 8 𝑥𝑚
7 nfcv 2761 . . . . . . . 8 𝑥 +
83nfcri 2755 . . . . . . . . . 10 𝑥 𝑛𝐴
9 nfcv 2761 . . . . . . . . . . 11 𝑥𝑛
10 nfsum.2 . . . . . . . . . . 11 𝑥𝐵
119, 10nfcsb 3532 . . . . . . . . . 10 𝑥𝑛 / 𝑘𝐵
12 nfcv 2761 . . . . . . . . . 10 𝑥0
138, 11, 12nfif 4087 . . . . . . . . 9 𝑥if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
142, 13nfmpt 4706 . . . . . . . 8 𝑥(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
156, 7, 14nfseq 12751 . . . . . . 7 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
16 nfcv 2761 . . . . . . 7 𝑥
17 nfcv 2761 . . . . . . 7 𝑥𝑧
1815, 16, 17nfbr 4659 . . . . . 6 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧
195, 18nfan 1825 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
202, 19nfrex 3001 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
21 nfcv 2761 . . . . 5 𝑥
22 nfcv 2761 . . . . . . . 8 𝑥𝑓
23 nfcv 2761 . . . . . . . 8 𝑥(1...𝑚)
2422, 23, 3nff1o 6092 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
25 nfcv 2761 . . . . . . . . . 10 𝑥1
26 nfcv 2761 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
2726, 10nfcsb 3532 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
2821, 27nfmpt 4706 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2925, 7, 28nfseq 12751 . . . . . . . . 9 𝑥seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3029, 6nffv 6155 . . . . . . . 8 𝑥(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3130nfeq2 2776 . . . . . . 7 𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3224, 31nfan 1825 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3332nfex 2151 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3421, 33nfrex 3001 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3520, 34nfor 1831 . . 3 𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
3635nfiota 5814 . 2 𝑥(℩𝑧(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
371, 36nfcxfr 2759 1 𝑥Σ𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wnfc 2748  wrex 2908  csb 3514  wss 3555  ifcif 4058   class class class wbr 4613  cmpt 4673  cio 5808  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  0cc0 9880  1c1 9881   + caddc 9883  cn 10964  cz 11321  cuz 11631  ...cfz 12268  seqcseq 12741  cli 14149  Σcsu 14350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-seq 12742  df-sum 14351
This theorem is referenced by:  fsum2dlem  14429  fsumcom2  14433  fsumcom2OLD  14434  fsumrlim  14470  fsumiun  14480  fsumcn  22581  fsum2cn  22582  nfitg1  23446  nfitg  23447  dvmptfsum  23642  fsumdvdscom  24811  binomcxplemdvsum  38033  binomcxplemnotnn0  38034  fsumcnf  38660  fsumiunss  39208  dvmptfprod  39463  sge0iunmptlemre  39936
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