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Theorem seqeq2 11332
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq2  |-  (  .+  =  Q  ->  seq  M
(  .+  ,  F
)  =  seq  M
( Q ,  F
) )

Proof of Theorem seqeq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2439 . . . . 5  |-  (  .+  =  Q  ->  _V  =  _V )
2 oveq 6090 . . . . . 6  |-  (  .+  =  Q  ->  ( y 
.+  ( F `  ( x  +  1
) ) )  =  ( y Q ( F `  ( x  +  1 ) ) ) )
32opeq2d 3993 . . . . 5  |-  (  .+  =  Q  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
41, 1, 3mpt2eq123dv 6139 . . . 4  |-  (  .+  =  Q  ->  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y Q ( F `  ( x  +  1
) ) ) >.
) )
5 rdgeq1 6672 . . . 4  |-  ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
)  =  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )  ->  rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  =  rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M )
>. ) )
64, 5syl 16 . . 3  |-  (  .+  =  Q  ->  rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  =  rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y Q ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) )
76imaeq1d 5205 . 2  |-  (  .+  =  Q  ->  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y Q ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om ) )
8 df-seq 11329 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
9 df-seq 11329 . 2  |-  seq  M
( Q ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y Q ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
107, 8, 93eqtr4g 2495 1  |-  (  .+  =  Q  ->  seq  M
(  .+  ,  F
)  =  seq  M
( Q ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   _Vcvv 2958   <.cop 3819   omcom 4848   "cima 4884   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   reccrdg 6670   1c1 8996    + caddc 8998    seq cseq 11328
This theorem is referenced by:  seqeq2d  11335  sadcom  12980  gxfval  21850  ressmulgnn  24210  cvmliftlem15  24990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-recs 6636  df-rdg 6671  df-seq 11329
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