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Theorem nfseq 10258
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 10249 . 2  |-  seq M
(  .+  ,  F
)  =  ran frec (
( z  e.  (
ZZ>= `  M ) ,  w  e.  _V  |->  <.
( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
2 nfcv 2282 . . . . . 6  |-  F/_ x ZZ>=
3 nfseq.1 . . . . . 6  |-  F/_ x M
42, 3nffv 5438 . . . . 5  |-  F/_ x
( ZZ>= `  M )
5 nfcv 2282 . . . . 5  |-  F/_ x _V
6 nfcv 2282 . . . . . 6  |-  F/_ x
( z  +  1 )
7 nfcv 2282 . . . . . . 7  |-  F/_ x w
8 nfseq.2 . . . . . . 7  |-  F/_ x  .+
9 nfseq.3 . . . . . . . 8  |-  F/_ x F
109, 6nffv 5438 . . . . . . 7  |-  F/_ x
( F `  (
z  +  1 ) )
117, 8, 10nfov 5808 . . . . . 6  |-  F/_ x
( w  .+  ( F `  ( z  +  1 ) ) )
126, 11nfop 3728 . . . . 5  |-  F/_ x <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
134, 5, 12nfmpo 5847 . . . 4  |-  F/_ x
( z  e.  (
ZZ>= `  M ) ,  w  e.  _V  |->  <.
( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
)
149, 3nffv 5438 . . . . 5  |-  F/_ x
( F `  M
)
153, 14nfop 3728 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1613, 15nffrec 6300 . . 3  |-  F/_ xfrec ( ( z  e.  ( ZZ>= `  M ) ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
1716nfrn 4791 . 2  |-  F/_ x ran frec ( ( z  e.  ( ZZ>= `  M ) ,  w  e.  _V  |->  <. ( z  +  1 ) ,  ( w 
.+  ( F `  ( z  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
181, 17nfcxfr 2279 1  |-  F/_ x  seq M (  .+  ,  F )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2269   _Vcvv 2689   <.cop 3534   ran crn 4547   ` cfv 5130  (class class class)co 5781    e. cmpo 5783  freccfrec 6294   1c1 7644    + caddc 7646   ZZ>=cuz 9349    seqcseq 10248
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-un 3079  df-in 3081  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-xp 4552  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-iota 5095  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-recs 6209  df-frec 6295  df-seqfrec 10249
This theorem is referenced by:  seq3f1olemstep  10304  seq3f1olemp  10305  nfsum1  11156  nfsum  11157  nfcprod1  11354  nfcprod  11355
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