MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfseq Structured version   Unicode version

Theorem nfseq 11338
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 11329 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
2 nfcv 2574 . . . . 5  |-  F/_ x _V
3 nfcv 2574 . . . . . 6  |-  F/_ x
( z  +  1 )
4 nfcv 2574 . . . . . . 7  |-  F/_ x w
5 nfseq.2 . . . . . . 7  |-  F/_ x  .+
6 nfseq.3 . . . . . . . 8  |-  F/_ x F
76, 3nffv 5738 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
84, 5, 7nfov 6107 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
93, 8nfop 4002 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
102, 2, 9nfmpt2 6145 . . . 4  |-  F/_ x
( z  e.  _V ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
11 nfseq.1 . . . . 5  |-  F/_ x M
126, 11nffv 5738 . . . . 5  |-  F/_ x
( F `  M
)
1311, 12nfop 4002 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1410, 13nfrdg 6675 . . 3  |-  F/_ x rec ( ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( w  .+  ( F `
 ( z  +  1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )
15 nfcv 2574 . . 3  |-  F/_ x om
1614, 15nfima 5214 . 2  |-  F/_ x
( rec ( ( z  e.  _V ,  w  e.  _V  |->  <. (
z  +  1 ) ,  ( w  .+  ( F `  ( z  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. ) " om )
171, 16nfcxfr 2571 1  |-  F/_ x  seq  M (  .+  ,  F )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2561   _Vcvv 2958   <.cop 3819   omcom 4848   "cima 4884   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   reccrdg 6670   1c1 8996    + caddc 8998    seq cseq 11328
This theorem is referenced by:  seqof2  11386  nfsum1  12489  nfsum  12490  lgamgulm2  24825  nfcprod1  25241  nfcprod  25242  fmuldfeqlem1  27702  fmuldfeq  27703  stoweidlem51  27790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-seq 11329
  Copyright terms: Public domain W3C validator