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Theorem seqeq3 11257
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq3  |-  ( F  =  G  ->  seq  M (  .+  ,  F
)  =  seq  M
(  .+  ,  G
) )

Proof of Theorem seqeq3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2390 . . . . 5  |-  ( F  =  G  ->  _V  =  _V )
2 fveq1 5669 . . . . . . 7  |-  ( F  =  G  ->  ( F `  ( x  +  1 ) )  =  ( G `  ( x  +  1
) ) )
32oveq2d 6038 . . . . . 6  |-  ( F  =  G  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( G `  ( x  +  1 ) ) ) )
43opeq2d 3935 . . . . 5  |-  ( F  =  G  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. )
51, 1, 4mpt2eq123dv 6077 . . . 4  |-  ( F  =  G  ->  (
x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
)  =  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. ) )
6 fveq1 5669 . . . . 5  |-  ( F  =  G  ->  ( F `  M )  =  ( G `  M ) )
76opeq2d 3935 . . . 4  |-  ( F  =  G  ->  <. M , 
( F `  M
) >.  =  <. M , 
( G `  M
) >. )
8 rdgeq12 6609 . . . 4  |-  ( ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
)  =  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. )  /\  <. M ,  ( F `  M ) >.  =  <. M ,  ( G `  M ) >. )  ->  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  .+  ( G `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( G `  M ) >. )
)
95, 7, 8syl2anc 643 . . 3  |-  ( F  =  G  ->  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  .+  ( G `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( G `  M
) >. ) )
109imaeq1d 5144 . 2  |-  ( F  =  G  ->  ( 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  .+  ( G `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( G `  M
) >. ) " om ) )
11 df-seq 11253 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
12 df-seq 11253 . 2  |-  seq  M
(  .+  ,  G
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( G `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( G `  M
) >. ) " om )
1310, 11, 123eqtr4g 2446 1  |-  ( F  =  G  ->  seq  M (  .+  ,  F
)  =  seq  M
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2901   <.cop 3762   omcom 4787   "cima 4823   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   reccrdg 6605   1c1 8926    + caddc 8928    seq cseq 11252
This theorem is referenced by:  seqeq3d  11260  geolim3  20125  leibpilem2  20650  basel  20741  cbvprod  25022  iprodmul  25090  faclim  25125  ovoliunnfl  25955  voliunnfl  25957  heiborlem10  26222
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-cnv 4828  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-recs 6571  df-rdg 6606  df-seq 11253
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