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Theorem seqeq2 11290
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 2413 . . . . 5  |-  (  .+  =  Q  ->  _V  =  _V )
2 oveq 6054 . . . . . 6  |-  (  .+  =  Q  ->  ( y 
.+  ( F `  ( x  +  1
) ) )  =  ( y Q ( F `  ( x  +  1 ) ) ) )
32opeq2d 3959 . . . . 5  |-  (  .+  =  Q  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
41, 1, 3mpt2eq123dv 6103 . . . 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 6636 . . . 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 5169 . 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 11287 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
9 df-seq 11287 . 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 2469 1  |-  (  .+  =  Q  ->  seq  M
(  .+  ,  F
)  =  seq  M
( Q ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2924   <.cop 3785   omcom 4812   "cima 4848   ` cfv 5421  (class class class)co 6048    e. cmpt2 6050   reccrdg 6634   1c1 8955    + caddc 8957    seq cseq 11286
This theorem is referenced by:  seqeq2d  11293  sadcom  12938  gxfval  21806  ressmulgnn  24166  cvmliftlem15  24946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-seq 11287
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