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Theorem nfseq 11292
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 11283 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
2 nfcv 2544 . . . . 5  |-  F/_ x _V
3 nfcv 2544 . . . . . 6  |-  F/_ x
( z  +  1 )
4 nfcv 2544 . . . . . . 7  |-  F/_ x w
5 nfseq.2 . . . . . . 7  |-  F/_ x  .+
6 nfseq.3 . . . . . . . 8  |-  F/_ x F
76, 3nffv 5698 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
84, 5, 7nfov 6067 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
93, 8nfop 3964 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
102, 2, 9nfmpt2 6105 . . . 4  |-  F/_ x
( z  e.  _V ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
11 nfseq.1 . . . . 5  |-  F/_ x M
126, 11nffv 5698 . . . . 5  |-  F/_ x
( F `  M
)
1311, 12nfop 3964 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1410, 13nfrdg 6635 . . 3  |-  F/_ x rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )
15 nfcv 2544 . . 3  |-  F/_ x om
1614, 15nfima 5174 . 2  |-  F/_ x
( rec ( ( z  e.  _V ,  w  e.  _V  |->  <. (
z  +  1 ) ,  ( w  .+  ( F `  ( z  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. ) " om )
171, 16nfcxfr 2541 1  |-  F/_ x  seq  M (  .+  ,  F )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2531   _Vcvv 2920   <.cop 3781   omcom 4808   "cima 4844   ` cfv 5417  (class class class)co 6044    e. cmpt2 6046   reccrdg 6630   1c1 8951    + caddc 8953    seq cseq 11282
This theorem is referenced by:  seqof2  11340  nfsum1  12443  nfsum  12444  lgamgulm2  24777  nfcprod1  25193  nfcprod  25194  fmuldfeqlem1  27583  fmuldfeq  27584  stoweidlem51  27671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-xp 4847  df-cnv 4849  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-recs 6596  df-rdg 6631  df-seq 11283
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