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Theorem nfseq 10469
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-seqfrec 10460 . 2  |-  seq M
(  .+  ,  F
)  =  ran frec (
( z  e.  (
ZZ>= `  M ) ,  w  e.  _V  |->  <.
( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
2 nfcv 2329 . . . . . 6  |-  F/_ x ZZ>=
3 nfseq.1 . . . . . 6  |-  F/_ x M
42, 3nffv 5537 . . . . 5  |-  F/_ x
( ZZ>= `  M )
5 nfcv 2329 . . . . 5  |-  F/_ x _V
6 nfcv 2329 . . . . . 6  |-  F/_ x
( z  +  1 )
7 nfcv 2329 . . . . . . 7  |-  F/_ x w
8 nfseq.2 . . . . . . 7  |-  F/_ x  .+
9 nfseq.3 . . . . . . . 8  |-  F/_ x F
109, 6nffv 5537 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
117, 8, 10nfov 5918 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
126, 11nfop 3806 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
134, 5, 12nfmpo 5957 . . . 4  |-  F/_ x
( z  e.  (
ZZ>= `  M ) ,  w  e.  _V  |->  <.
( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
149, 3nffv 5537 . . . . 5  |-  F/_ x
( F `  M
)
153, 14nfop 3806 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1613, 15nffrec 6411 . . 3  |-  F/_ xfrec ( ( z  e.  ( ZZ>= `  M ) ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
1716nfrn 4884 . 2  |-  F/_ x ran frec ( ( z  e.  ( ZZ>= `  M ) ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
181, 17nfcxfr 2326 1  |-  F/_ x  seq M (  .+  ,  F )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2316   _Vcvv 2749   <.cop 3607   ran crn 4639   ` cfv 5228  (class class class)co 5888    e. cmpo 5890  freccfrec 6405   1c1 7826    + caddc 7828   ZZ>=cuz 9542    seqcseq 10459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-un 3145  df-in 3147  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-mpt 4078  df-xp 4644  df-cnv 4646  df-dm 4648  df-rn 4649  df-res 4650  df-iota 5190  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-recs 6320  df-frec 6406  df-seqfrec 10460
This theorem is referenced by:  seq3f1olemstep  10515  seq3f1olemp  10516  nfsum1  11378  nfsum  11379  nfcprod1  11576  nfcprod  11577
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