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Theorem seqeq3 11320
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 2436 . . . . 5  |-  ( F  =  G  ->  _V  =  _V )
2 fveq1 5719 . . . . . . 7  |-  ( F  =  G  ->  ( F `  ( x  +  1 ) )  =  ( G `  ( x  +  1
) ) )
32oveq2d 6089 . . . . . 6  |-  ( F  =  G  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( G `  ( x  +  1 ) ) ) )
43opeq2d 3983 . . . . 5  |-  ( F  =  G  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. )
51, 1, 4mpt2eq123dv 6128 . . . 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 5719 . . . . 5  |-  ( F  =  G  ->  ( F `  M )  =  ( G `  M ) )
76opeq2d 3983 . . . 4  |-  ( F  =  G  ->  <. M , 
( F `  M
) >.  =  <. M , 
( G `  M
) >. )
8 rdgeq12 6663 . . . 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 5194 . 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 11316 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
12 df-seq 11316 . 2  |-  seq  M
(  .+  ,  G
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( G `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( G `  M
) >. ) " om )
1310, 11, 123eqtr4g 2492 1  |-  ( F  =  G  ->  seq  M (  .+  ,  F
)  =  seq  M
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   _Vcvv 2948   <.cop 3809   omcom 4837   "cima 4873   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   reccrdg 6659   1c1 8983    + caddc 8985    seq cseq 11315
This theorem is referenced by:  seqeq3d  11323  geolim3  20248  leibpilem2  20773  basel  20864  cbvprod  25233  iprodmul  25308  faclim  25357  ovoliunnfl  26238  voliunnfl  26240  heiborlem10  26520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-seq 11316
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