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Theorem nfseq 11928
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 11919 . 2  |-  seq M
(  .+  ,  F
)  =  ( rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
2 nfcv 2614 . . . . 5  |-  F/_ x _V
3 nfcv 2614 . . . . . 6  |-  F/_ x
( z  +  1 )
4 nfcv 2614 . . . . . . 7  |-  F/_ x w
5 nfseq.2 . . . . . . 7  |-  F/_ x  .+
6 nfseq.3 . . . . . . . 8  |-  F/_ x F
76, 3nffv 5801 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
84, 5, 7nfov 6218 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
93, 8nfop 4178 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
102, 2, 9nfmpt2 6259 . . . 4  |-  F/_ x
( z  e.  _V ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
11 nfseq.1 . . . . 5  |-  F/_ x M
126, 11nffv 5801 . . . . 5  |-  F/_ x
( F `  M
)
1311, 12nfop 4178 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1410, 13nfrdg 6975 . . 3  |-  F/_ x rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )
15 nfcv 2614 . . 3  |-  F/_ x om
1614, 15nfima 5280 . 2  |-  F/_ x
( rec ( ( z  e.  _V ,  w  e.  _V  |->  <. (
z  +  1 ) ,  ( w  .+  ( F `  ( z  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. ) " om )
171, 16nfcxfr 2612 1  |-  F/_ x  seq M (  .+  ,  F )
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2600   _Vcvv 3072   <.cop 3986   "cima 4946   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   omcom 6581   reccrdg 6970   1c1 9389    + caddc 9391    seqcseq 11918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-xp 4949  df-cnv 4951  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-recs 6937  df-rdg 6971  df-seq 11919
This theorem is referenced by:  seqof2  11976  nfsum1  13280  nfsum  13281  lgamgulm2  27161  nfcprod1  27562  nfcprod  27563  fmuldfeqlem1  29906  fmuldfeq  29907  stoweidlem51  29989
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