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Theorem nfseq 10196
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 10187 . 2  |-  seq M
(  .+  ,  F
)  =  ran frec (
( z  e.  (
ZZ>= `  M ) ,  w  e.  _V  |->  <.
( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
2 nfcv 2258 . . . . . 6  |-  F/_ x ZZ>=
3 nfseq.1 . . . . . 6  |-  F/_ x M
42, 3nffv 5399 . . . . 5  |-  F/_ x
( ZZ>= `  M )
5 nfcv 2258 . . . . 5  |-  F/_ x _V
6 nfcv 2258 . . . . . 6  |-  F/_ x
( z  +  1 )
7 nfcv 2258 . . . . . . 7  |-  F/_ x w
8 nfseq.2 . . . . . . 7  |-  F/_ x  .+
9 nfseq.3 . . . . . . . 8  |-  F/_ x F
109, 6nffv 5399 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
117, 8, 10nfov 5769 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
126, 11nfop 3691 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
134, 5, 12nfmpo 5808 . . . 4  |-  F/_ x
( z  e.  (
ZZ>= `  M ) ,  w  e.  _V  |->  <.
( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
149, 3nffv 5399 . . . . 5  |-  F/_ x
( F `  M
)
153, 14nfop 3691 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1613, 15nffrec 6261 . . 3  |-  F/_ xfrec ( ( z  e.  ( ZZ>= `  M ) ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
1716nfrn 4754 . 2  |-  F/_ x ran frec ( ( z  e.  ( ZZ>= `  M ) ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
181, 17nfcxfr 2255 1  |-  F/_ x  seq M (  .+  ,  F )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2245   _Vcvv 2660   <.cop 3500   ran crn 4510   ` cfv 5093  (class class class)co 5742    e. cmpo 5744  freccfrec 6255   1c1 7589    + caddc 7591   ZZ>=cuz 9294    seqcseq 10186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-un 3045  df-in 3047  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-xp 4515  df-cnv 4517  df-dm 4519  df-rn 4520  df-res 4521  df-iota 5058  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-recs 6170  df-frec 6256  df-seqfrec 10187
This theorem is referenced by:  seq3f1olemstep  10242  seq3f1olemp  10243  nfsum1  11093  nfsum  11094
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