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Theorem seqeq2 11142
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 2359 . . . . 5  |-  (  .+  =  Q  ->  _V  =  _V )
2 oveq 5951 . . . . . 6  |-  (  .+  =  Q  ->  ( y 
.+  ( F `  ( x  +  1
) ) )  =  ( y Q ( F `  ( x  +  1 ) ) ) )
32opeq2d 3884 . . . . 5  |-  (  .+  =  Q  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
41, 1, 3mpt2eq123dv 5997 . . . 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 6511 . . . 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 15 . . 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 5093 . 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 11139 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
9 df-seq 11139 . 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 2415 1  |-  (  .+  =  Q  ->  seq  M
(  .+  ,  F
)  =  seq  M
( Q ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642   _Vcvv 2864   <.cop 3719   omcom 4738   "cima 4774   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   reccrdg 6509   1c1 8828    + caddc 8830    seq cseq 11138
This theorem is referenced by:  seqeq2d  11145  sadcom  12751  gxfval  21036  ressmulgnn  23397  cvmliftlem15  24233
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-recs 6475  df-rdg 6510  df-seq 11139
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