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Theorem nfiseq 9789
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 9783 . 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 2225 . . . . . 6  |-  F/_ x ZZ>=
3 nfiseq.1 . . . . . 6  |-  F/_ x M
42, 3nffv 5280 . . . . 5  |-  F/_ x
( ZZ>= `  M )
5 nfiseq.4 . . . . 5  |-  F/_ x S
6 nfcv 2225 . . . . . 6  |-  F/_ x
( y  +  1 )
7 nfcv 2225 . . . . . . 7  |-  F/_ x
z
8 nfiseq.2 . . . . . . 7  |-  F/_ x  .+
9 nfiseq.3 . . . . . . . 8  |-  F/_ x F
109, 6nffv 5280 . . . . . . 7  |-  F/_ x
( F `  (
y  +  1 ) )
117, 8, 10nfov 5638 . . . . . 6  |-  F/_ x
( z  .+  ( F `  ( y  +  1 ) ) )
126, 11nfop 3623 . . . . 5  |-  F/_ x <. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
134, 5, 12nfmpt2 5676 . . . 4  |-  F/_ x
( y  e.  (
ZZ>= `  M ) ,  z  e.  S  |->  <.
( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
)
149, 3nffv 5280 . . . . 5  |-  F/_ x
( F `  M
)
153, 14nfop 3623 . . . 4  |-  F/_ x <. M ,  ( F `
 M ) >.
1613, 15nffrec 6117 . . 3  |-  F/_ xfrec ( ( y  e.  ( ZZ>= `  M ) ,  z  e.  S  |-> 
<. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
1716nfrn 4650 . 2  |-  F/_ x ran frec ( ( y  e.  ( ZZ>= `  M ) ,  z  e.  S  |-> 
<. ( y  +  1 ) ,  ( z 
.+  ( F `  ( y  +  1 ) ) ) >.
) ,  <. M , 
( F `  M
) >. )
181, 17nfcxfr 2222 1  |-  F/_ x  seq M (  .+  ,  F ,  S )
Colors of variables: wff set class
Syntax hints:   F/_wnfc 2212   <.cop 3434   ran crn 4414   ` cfv 4983  (class class class)co 5615    |-> cmpt2 5617  freccfrec 6111   1c1 7298    + caddc 7300   ZZ>=cuz 8954    seqcseq 9782
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-rab 2364  df-v 2617  df-un 2992  df-in 2994  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-br 3823  df-opab 3877  df-mpt 3878  df-xp 4419  df-cnv 4421  df-dm 4423  df-rn 4424  df-res 4425  df-iota 4948  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-recs 6026  df-frec 6112  df-iseq 9783
This theorem is referenced by:  iseqf1olemstep  9838  iseqf1olemp  9839  nfsum1  10640  nfsum  10641
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