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Theorem seqeq2 11913
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 2452 . . . . 5  |-  (  .+  =  Q  ->  _V  =  _V )
2 oveq 6198 . . . . . 6  |-  (  .+  =  Q  ->  ( y 
.+  ( F `  ( x  +  1
) ) )  =  ( y Q ( F `  ( x  +  1 ) ) ) )
32opeq2d 4166 . . . . 5  |-  (  .+  =  Q  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
41, 1, 3mpt2eq123dv 6249 . . . 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 6969 . . . 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 5268 . 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 11910 . 2  |-  seq M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
9 df-seq 11910 . 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 2517 1  |-  (  .+  =  Q  ->  seq M
(  .+  ,  F
)  =  seq M
( Q ,  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   _Vcvv 3070   <.cop 3983   "cima 4943   ` cfv 5518  (class class class)co 6192    |-> cmpt2 6194   omcom 6578   reccrdg 6967   1c1 9386    + caddc 9388    seqcseq 11909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fv 5526  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-recs 6934  df-rdg 6968  df-seq 11910
This theorem is referenced by:  seqeq2d  11916  sadcom  13763  gxfval  23881  ressmulgnn  26280  cvmliftlem15  27323
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