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Theorem nfseq 12099
Description: Hypothesis builder for the sequence builder operation. (Contributed by Mario Carneiro, 24-Jun-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
nfseq.1  |-  F/_ x M
nfseq.2  |-  F/_ x  .+
nfseq.3  |-  F/_ x F
Assertion
Ref Expression
nfseq  |-  F/_ x  seq M (  .+  ,  F )

Proof of Theorem nfseq
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-seq 12090 . 2  |-  seq M
(  .+  ,  F
)  =  ( rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
2 nfcv 2616 . . . . 5  |-  F/_ x _V
3 nfcv 2616 . . . . . 6  |-  F/_ x
( z  +  1 )
4 nfcv 2616 . . . . . . 7  |-  F/_ x w
5 nfseq.2 . . . . . . 7  |-  F/_ x  .+
6 nfseq.3 . . . . . . . 8  |-  F/_ x F
76, 3nffv 5855 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
84, 5, 7nfov 6296 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
93, 8nfop 4219 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
102, 2, 9nfmpt2 6339 . . . 4  |-  F/_ x
( z  e.  _V ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
11 nfseq.1 . . . . 5  |-  F/_ x M
126, 11nffv 5855 . . . . 5  |-  F/_ x
( F `  M
)
1311, 12nfop 4219 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1410, 13nfrdg 7072 . . 3  |-  F/_ x rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )
15 nfcv 2616 . . 3  |-  F/_ x om
1614, 15nfima 5333 . 2  |-  F/_ x
( rec ( ( z  e.  _V ,  w  e.  _V  |->  <. (
z  +  1 ) ,  ( w  .+  ( F `  ( z  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. ) " om )
171, 16nfcxfr 2614 1  |-  F/_ x  seq M (  .+  ,  F )
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2602   _Vcvv 3106   <.cop 4022   "cima 4991   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   omcom 6673   reccrdg 7067   1c1 9482    + caddc 9484    seqcseq 12089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-xp 4994  df-cnv 4996  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-recs 7034  df-rdg 7068  df-seq 12090
This theorem is referenced by:  seqof2  12147  nfsum1  13594  nfsum  13595  nfcprod1  13799  nfcprod  13800  lgamgulm2  28842  binomcxplemdvbinom  31499  binomcxplemdvsum  31501  binomcxplemnotnn0  31502  fmuldfeqlem1  31815  fmuldfeq  31816  sumnnodd  31875  stoweidlem51  32072
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