MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfsum Structured version   Visualization version   GIF version

Theorem nfsum 14269
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 14265 . 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 2751 . . . . 5 𝑥
3 nfsum.1 . . . . . . 7 𝑥𝐴
4 nfcv 2751 . . . . . . 7 𝑥(ℤ𝑚)
53, 4nfss 3561 . . . . . 6 𝑥 𝐴 ⊆ (ℤ𝑚)
6 nfcv 2751 . . . . . . . 8 𝑥𝑚
7 nfcv 2751 . . . . . . . 8 𝑥 +
83nfcri 2745 . . . . . . . . . 10 𝑥 𝑛𝐴
9 nfcv 2751 . . . . . . . . . . 11 𝑥𝑛
10 nfsum.2 . . . . . . . . . . 11 𝑥𝐵
119, 10nfcsb 3517 . . . . . . . . . 10 𝑥𝑛 / 𝑘𝐵
12 nfcv 2751 . . . . . . . . . 10 𝑥0
138, 11, 12nfif 4065 . . . . . . . . 9 𝑥if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)
142, 13nfmpt 4674 . . . . . . . 8 𝑥(𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))
156, 7, 14nfseq 12673 . . . . . . 7 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)))
16 nfcv 2751 . . . . . . 7 𝑥
17 nfcv 2751 . . . . . . 7 𝑥𝑧
1815, 16, 17nfbr 4629 . . . . . 6 𝑥seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧
195, 18nfan 1816 . . . . 5 𝑥(𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
202, 19nfrex 2990 . . . 4 𝑥𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0))) ⇝ 𝑧)
21 nfcv 2751 . . . . 5 𝑥
22 nfcv 2751 . . . . . . . 8 𝑥𝑓
23 nfcv 2751 . . . . . . . 8 𝑥(1...𝑚)
2422, 23, 3nff1o 6048 . . . . . . 7 𝑥 𝑓:(1...𝑚)–1-1-onto𝐴
25 nfcv 2751 . . . . . . . . . 10 𝑥1
26 nfcv 2751 . . . . . . . . . . . 12 𝑥(𝑓𝑛)
2726, 10nfcsb 3517 . . . . . . . . . . 11 𝑥(𝑓𝑛) / 𝑘𝐵
2821, 27nfmpt 4674 . . . . . . . . . 10 𝑥(𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
2925, 7, 28nfseq 12673 . . . . . . . . 9 𝑥seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))
3029, 6nffv 6110 . . . . . . . 8 𝑥(seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3130nfeq2 2766 . . . . . . 7 𝑥 𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)
3224, 31nfan 1816 . . . . . 6 𝑥(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3332nfex 2140 . . . . 5 𝑥𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3421, 33nfrex 2990 . . . 4 𝑥𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑧 = (seq1( + , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))
3520, 34nfor 1822 . . 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 5772 . 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 2749 1 𝑥Σ𝑘𝐴 𝐵
Colors of variables: wff setvar class
Syntax hints:  wo 382  wa 383   = wceq 1475  wex 1695  wcel 1977  wnfc 2738  wrex 2897  csb 3499  wss 3540  ifcif 4036   class class class wbr 4583  cmpt 4643  cio 5766  1-1-ontowf1o 5803  cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  cn 10897  cz 11254  cuz 11563  ...cfz 12197  seqcseq 12663  cli 14063  Σcsu 14264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-seq 12664  df-sum 14265
This theorem is referenced by:  fsum2dlem  14343  fsumcom2  14347  fsumcom2OLD  14348  fsumrlim  14384  fsumiun  14394  fsumcn  22481  fsum2cn  22482  nfitg1  23346  nfitg  23347  dvmptfsum  23542  fsumdvdscom  24711  binomcxplemdvsum  37576  binomcxplemnotnn0  37577  fsumcnf  38203  fsumiunss  38642  dvmptfprod  38835  sge0iunmptlemre  39308
  Copyright terms: Public domain W3C validator