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Theorem nfsum 15637
Description: Bound-variable hypothesis builder for sum: if 𝑥 is (effectively) not free in 𝐴 and 𝐵, it is not free in Σ𝑘𝐴𝐵. Version of nfsum 15637 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 11-Dec-2005.) (Revised by Gino Giotto, 24-Feb-2024.)
Hypotheses
Ref Expression
nfsum.1 𝑥𝐴
nfsum.2 𝑥𝐵
Assertion
Ref Expression
nfsum 𝑥Σ𝑘𝐴 𝐵
Distinct variable group:   𝑥,𝑘
Allowed substitution hints:   𝐴(𝑥,𝑘)   𝐵(𝑥,𝑘)

Proof of Theorem nfsum
Dummy variables 𝑓 𝑚 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-sum 15633 . 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 2904 . . . . 5 𝑥
3 nfsum.1 . . . . . . 7 𝑥𝐴
4 nfcv 2904 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3975 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2904 . . . . . . . 8 𝑥𝑚
7 nfcv 2904 . . . . . . . 8 𝑥 +
83nfcri 2891 . . . . . . . . . 10 𝑥 𝑛𝐴
9 nfcv 2904 . . . . . . . . . . 11 𝑥𝑛
10 nfsum.2 . . . . . . . . . . 11 𝑥𝐵
119, 10nfcsbw 3921 . . . . . . . . . 10 𝑥𝑛 / 𝑘𝐵
12 nfcv 2904 . . . . . . . . . 10 𝑥0
138, 11, 12nfif 4559 . . . . . . . . 9 𝑥if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
142, 13nfmpt 5256 . . . . . . . 8 𝑥(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
156, 7, 14nfseq 13976 . . . . . . 7 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
16 nfcv 2904 . . . . . . 7 𝑥
17 nfcv 2904 . . . . . . 7 𝑥𝑧
1815, 16, 17nfbr 5196 . . . . . 6 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧
195, 18nfan 1903 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
202, 19nfrexw 3311 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
21 nfcv 2904 . . . . 5 𝑥
22 nfcv 2904 . . . . . . . 8 𝑥𝑓
23 nfcv 2904 . . . . . . . 8 𝑥(1...𝑚)
2422, 23, 3nff1o 6832 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
25 nfcv 2904 . . . . . . . . . 10 𝑥1
26 nfcv 2904 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
2726, 10nfcsbw 3921 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
2821, 27nfmpt 5256 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2925, 7, 28nfseq 13976 . . . . . . . . 9 𝑥seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3029, 6nffv 6902 . . . . . . . 8 𝑥(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3130nfeq2 2921 . . . . . . 7 𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3224, 31nfan 1903 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3332nfex 2318 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3421, 33nfrexw 3311 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3520, 34nfor 1908 . . 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)))
3635nfiotaw 6500 . 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 2902 1 𝑥Σ𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wa 397  wo 846   = wceq 1542  wex 1782  wcel 2107  wnfc 2884  wrex 3071  csb 3894  wss 3949  ifcif 4529   class class class wbr 5149  cmpt 5232  cio 6494  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113  cn 12212  cz 12558  cuz 12822  ...cfz 13484  seqcseq 13966  cli 15428  Σcsu 15632
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 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-seq 13967  df-sum 15633
This theorem is referenced by:  fsum2dlem  15716  fsumcom2  15720  fsumrlim  15757  fsumiun  15767  fsumcn  24386  fsum2cn  24387  nfitg1  25291  nfitg  25292  dvmptfsum  25492  fsumdvdscom  26689  binomcxplemdvsum  43114  binomcxplemnotnn0  43115  fsumcnf  43705  fsumiunss  44291  dvmptfprod  44661  sge0iunmptlemre  45131
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