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Theorem nfiseq 9747
Description: Hypothesis builder for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Hypotheses
Ref Expression
nfiseq.1  |-  F/_ x M
nfiseq.2  |-  F/_ x  .+
nfiseq.3  |-  F/_ x F
nfiseq.4  |-  F/_ x S
Assertion
Ref Expression
nfiseq  |-  F/_ x  seq M (  .+  ,  F ,  S )

Proof of Theorem nfiseq
Dummy variables  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iseq 9741 . 2  |-  seq M
(  .+  ,  F ,  S )  =  ran frec ( ( y  e.  (
ZZ>= `  M ) ,  z  e.  S  |->  <.
( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
2 nfcv 2223 . . . . . 6  |-  F/_ x ZZ>=
3 nfiseq.1 . . . . . 6  |-  F/_ x M
42, 3nffv 5260 . . . . 5  |-  F/_ x
( ZZ>= `  M )
5 nfiseq.4 . . . . 5  |-  F/_ x S
6 nfcv 2223 . . . . . 6  |-  F/_ x
( y  +  1 )
7 nfcv 2223 . . . . . . 7  |-  F/_ x
z
8 nfiseq.2 . . . . . . 7  |-  F/_ x  .+
9 nfiseq.3 . . . . . . . 8  |-  F/_ x F
109, 6nffv 5260 . . . . . . 7  |-  F/_ x
( F `  (
y  +  1 ) )
117, 8, 10nfov 5614 . . . . . 6  |-  F/_ x
( z  .+  ( F `  ( y  +  1 ) ) )
126, 11nfop 3612 . . . . 5  |-  F/_ x <. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
134, 5, 12nfmpt2 5652 . . . 4  |-  F/_ x
( y  e.  (
ZZ>= `  M ) ,  z  e.  S  |->  <.
( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
)
149, 3nffv 5260 . . . . 5  |-  F/_ x
( F `  M
)
153, 14nfop 3612 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1613, 15nffrec 6093 . . 3  |-  F/_ xfrec ( ( y  e.  ( ZZ>= `  M ) ,  z  e.  S  |-> 
<. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
1716nfrn 4638 . 2  |-  F/_ x ran frec ( ( y  e.  ( ZZ>= `  M ) ,  z  e.  S  |-> 
<. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
181, 17nfcxfr 2220 1  |-  F/_ x  seq M (  .+  ,  F ,  S )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2210   <.cop 3425   ran crn 4402   ` cfv 4969  (class class class)co 5591    |-> cmpt2 5593  freccfrec 6087   1c1 7254    + caddc 7256   ZZ>=cuz 8914    seqcseq 9740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-un 2988  df-in 2990  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-br 3812  df-opab 3866  df-mpt 3867  df-xp 4407  df-cnv 4409  df-dm 4411  df-rn 4412  df-res 4413  df-iota 4934  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-recs 6002  df-frec 6088  df-iseq 9741
This theorem is referenced by:  nfsum1  10567  nfsum  10568
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