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Theorem nfseq 12085
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 12076 . 2  |-  seq M
(  .+  ,  F
)  =  ( rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
2 nfcv 2629 . . . . 5  |-  F/_ x _V
3 nfcv 2629 . . . . . 6  |-  F/_ x
( z  +  1 )
4 nfcv 2629 . . . . . . 7  |-  F/_ x w
5 nfseq.2 . . . . . . 7  |-  F/_ x  .+
6 nfseq.3 . . . . . . . 8  |-  F/_ x F
76, 3nffv 5873 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
84, 5, 7nfov 6307 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
93, 8nfop 4229 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
102, 2, 9nfmpt2 6350 . . . 4  |-  F/_ x
( z  e.  _V ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
11 nfseq.1 . . . . 5  |-  F/_ x M
126, 11nffv 5873 . . . . 5  |-  F/_ x
( F `  M
)
1311, 12nfop 4229 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1410, 13nfrdg 7080 . . 3  |-  F/_ x rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )
15 nfcv 2629 . . 3  |-  F/_ x om
1614, 15nfima 5345 . 2  |-  F/_ x
( rec ( ( z  e.  _V ,  w  e.  _V  |->  <. (
z  +  1 ) ,  ( w  .+  ( F `  ( z  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. ) " om )
171, 16nfcxfr 2627 1  |-  F/_ x  seq M (  .+  ,  F )
Colors of variables: wff setvar class
Syntax hints:   F/_wnfc 2615   _Vcvv 3113   <.cop 4033   "cima 5002   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   omcom 6684   reccrdg 7075   1c1 9493    + caddc 9495    seqcseq 12075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-recs 7042  df-rdg 7076  df-seq 12076
This theorem is referenced by:  seqof2  12133  nfsum1  13475  nfsum  13476  lgamgulm2  28246  nfcprod1  28647  nfcprod  28648  fmuldfeqlem1  31160  fmuldfeq  31161  sumnnodd  31200  stoweidlem51  31379
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