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Theorem seqeq3 11067
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 2297 . . . . 5  |-  ( F  =  G  ->  _V  =  _V )
2 fveq1 5540 . . . . . . 7  |-  ( F  =  G  ->  ( F `  ( x  +  1 ) )  =  ( G `  ( x  +  1
) ) )
32oveq2d 5890 . . . . . 6  |-  ( F  =  G  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( G `  ( x  +  1 ) ) ) )
43opeq2d 3819 . . . . 5  |-  ( F  =  G  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. )
51, 1, 4mpt2eq123dv 5926 . . . 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 5540 . . . . 5  |-  ( F  =  G  ->  ( F `  M )  =  ( G `  M ) )
76opeq2d 3819 . . . 4  |-  ( F  =  G  ->  <. M , 
( F `  M
) >.  =  <. M , 
( G `  M
) >. )
8 rdgeq12 6442 . . . 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 642 . . 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 5027 . 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 11063 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
12 df-seq 11063 . 2  |-  seq  M
(  .+  ,  G
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( G `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( G `  M
) >. ) " om )
1310, 11, 123eqtr4g 2353 1  |-  ( F  =  G  ->  seq  M (  .+  ,  F
)  =  seq  M
(  .+  ,  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632   _Vcvv 2801   <.cop 3656   omcom 4672   "cima 4708   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   reccrdg 6438   1c1 8754    + caddc 8756    seq cseq 11062
This theorem is referenced by:  seqeq3d  11070  geolim3  19735  leibpilem2  20253  basel  20343  esumpcvgval  23461  faclimlem3  24119  faclim  24126  cbvcprod  24137  ovoliunnfl  25001  cbvprodi  25415  heiborlem10  26647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-seq 11063
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