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Theorem nfsum1 14464
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 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 14461 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
2 nfcv 2793 . . . . 5 𝑘
3 nfsum1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2793 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 3629 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2793 . . . . . . . 8 𝑘𝑚
7 nfcv 2793 . . . . . . . 8 𝑘 +
83nfcri 2787 . . . . . . . . . 10 𝑘 𝑛𝐴
9 nfcsb1v 3582 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
10 nfcv 2793 . . . . . . . . . 10 𝑘0
118, 9, 10nfif 4148 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
122, 11nfmpt 4779 . . . . . . . 8 𝑘(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
136, 7, 12nfseq 12851 . . . . . . 7 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
14 nfcv 2793 . . . . . . 7 𝑘
15 nfcv 2793 . . . . . . 7 𝑘𝑥
1613, 14, 15nfbr 4732 . . . . . 6 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
175, 16nfan 1868 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
182, 17nfrex 3036 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
19 nfcv 2793 . . . . 5 𝑘
20 nfcv 2793 . . . . . . . 8 𝑘𝑓
21 nfcv 2793 . . . . . . . 8 𝑘(1...𝑚)
2220, 21, 3nff1o 6173 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
23 nfcv 2793 . . . . . . . . . 10 𝑘1
24 nfcsb1v 3582 . . . . . . . . . . 11 𝑘(𝑓𝑛) / 𝑘𝐵
2519, 24nfmpt 4779 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2623, 7, 25nfseq 12851 . . . . . . . . 9 𝑘seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
2726, 6nffv 6236 . . . . . . . 8 𝑘(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
2827nfeq2 2809 . . . . . . 7 𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
2922, 28nfan 1868 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3029nfex 2192 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3119, 30nfrex 3036 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3218, 31nfor 1874 . . 3 𝑘(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))
3332nfiota 5893 . 2 𝑘(℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
341, 33nfcxfr 2791 1 𝑘Σ𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wo 382  wa 383   = wceq 1523  wex 1744  wcel 2030  wnfc 2780  wrex 2942  csb 3566  wss 3607  ifcif 4119   class class class wbr 4685  cmpt 4762  cio 5887  1-1-ontowf1o 5925  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cn 11058  cz 11415  cuz 11725  ...cfz 12364  seqcseq 12841  cli 14259  Σcsu 14460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-seq 12842  df-sum 14461
This theorem is referenced by:  dvmptfprod  40478  dvnprodlem1  40479  fourierdlem112  40753  etransclem32  40801  sge0reuz  40982
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