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Theorem nfsum1 15580
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 15577 . 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 2904 . . . . 5 𝑘
3 nfsum1.1 . . . . . . 7 𝑘𝐴
4 nfcv 2904 . . . . . . 7 𝑘(ℤ𝑚)
53, 4nfss 3937 . . . . . 6 𝑘 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2904 . . . . . . . 8 𝑘𝑚
7 nfcv 2904 . . . . . . . 8 𝑘 +
83nfcri 2891 . . . . . . . . . 10 𝑘 𝑛𝐴
9 nfcsb1v 3881 . . . . . . . . . 10 𝑘𝑛 / 𝑘𝐵
10 nfcv 2904 . . . . . . . . . 10 𝑘0
118, 9, 10nfif 4517 . . . . . . . . 9 𝑘if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
122, 11nfmpt 5213 . . . . . . . 8 𝑘(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
136, 7, 12nfseq 13922 . . . . . . 7 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
14 nfcv 2904 . . . . . . 7 𝑘
15 nfcv 2904 . . . . . . 7 𝑘𝑥
1613, 14, 15nfbr 5153 . . . . . 6 𝑘seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥
175, 16nfan 1903 . . . . 5 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
182, 17nfrexw 3295 . . . 4 𝑘𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑥)
19 nfcv 2904 . . . . 5 𝑘
20 nfcv 2904 . . . . . . . 8 𝑘𝑓
21 nfcv 2904 . . . . . . . 8 𝑘(1...𝑚)
2220, 21, 3nff1o 6783 . . . . . . 7 𝑘 𝑓:(1...𝑚)–1-1-onto𝐴
23 nfcv 2904 . . . . . . . . . 10 𝑘1
24 nfcsb1v 3881 . . . . . . . . . . 11 𝑘(𝑓𝑛) / 𝑘𝐵
2519, 24nfmpt 5213 . . . . . . . . . 10 𝑘(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2623, 7, 25nfseq 13922 . . . . . . . . 9 𝑘seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
2726, 6nffv 6853 . . . . . . . 8 𝑘(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
2827nfeq2 2921 . . . . . . 7 𝑘 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
2922, 28nfan 1903 . . . . . 6 𝑘(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3029nfex 2318 . . . . 5 𝑘𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3119, 30nfrexw 3295 . . . 4 𝑘𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3218, 31nfor 1908 . . 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)))
3332nfiotaw 6453 . 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 2902 1 𝑘Σ𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wnfc 2884  wrex 3070  csb 3856  wss 3911  ifcif 4487   class class class wbr 5106  cmpt 5189  cio 6447  1-1-ontowf1o 6496  cfv 6497  (class class class)co 7358  0cc0 11056  1c1 11057   + caddc 11059  cn 12158  cz 12504  cuz 12768  ...cfz 13430  seqcseq 13912  cli 15372  Σcsu 15576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-seq 13913  df-sum 15577
This theorem is referenced by:  dvmptfprod  44272  dvnprodlem1  44273  fourierdlem112  44545  etransclem32  44593  sge0reuz  44774
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