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Theorem nfiseq 9218
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 9212 . 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 2178 . . . . . 6  |-  F/_ x ZZ>=
3 nfiseq.1 . . . . . 6  |-  F/_ x M
42, 3nffv 5185 . . . . 5  |-  F/_ x
( ZZ>= `  M )
5 nfiseq.4 . . . . 5  |-  F/_ x S
6 nfcv 2178 . . . . . 6  |-  F/_ x
( y  +  1 )
7 nfcv 2178 . . . . . . 7  |-  F/_ x
z
8 nfiseq.2 . . . . . . 7  |-  F/_ x  .+
9 nfiseq.3 . . . . . . . 8  |-  F/_ x F
109, 6nffv 5185 . . . . . . 7  |-  F/_ x
( F `  (
y  +  1 ) )
117, 8, 10nfov 5535 . . . . . 6  |-  F/_ x
( z  .+  ( F `  ( y  +  1 ) ) )
126, 11nfop 3565 . . . . 5  |-  F/_ x <. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
134, 5, 12nfmpt2 5573 . . . 4  |-  F/_ x
( y  e.  (
ZZ>= `  M ) ,  z  e.  S  |->  <.
( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
)
149, 3nffv 5185 . . . . 5  |-  F/_ x
( F `  M
)
153, 14nfop 3565 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1613, 15nffrec 5982 . . 3  |-  F/_ xfrec ( ( y  e.  ( ZZ>= `  M ) ,  z  e.  S  |-> 
<. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
1716nfrn 4579 . 2  |-  F/_ x ran frec ( ( y  e.  ( ZZ>= `  M ) ,  z  e.  S  |-> 
<. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
181, 17nfcxfr 2175 1  |-  F/_ x  seq M (  .+  ,  F ,  S )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2165   <.cop 3378   ran crn 4346   ` cfv 4902  (class class class)co 5512    |-> cmpt2 5514  freccfrec 5977   1c1 6890    + caddc 6892   ZZ>=cuz 8473    seqcseq 9211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-un 2922  df-in 2924  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-iota 4867  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-recs 5920  df-frec 5978  df-iseq 9212
This theorem is referenced by:  nfsum1  9875  nfsum  9876
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