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Theorem seqeq1 11328
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq1  |-  ( M  =  N  ->  seq  M (  .+  ,  F
)  =  seq  N
(  .+  ,  F
) )

Proof of Theorem seqeq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5730 . . . . 5  |-  ( M  =  N  ->  ( F `  M )  =  ( F `  N ) )
2 opeq12 3988 . . . . 5  |-  ( ( M  =  N  /\  ( F `  M )  =  ( F `  N ) )  ->  <. M ,  ( F `
 M ) >.  =  <. N ,  ( F `  N )
>. )
31, 2mpdan 651 . . . 4  |-  ( M  =  N  ->  <. M , 
( F `  M
) >.  =  <. N , 
( F `  N
) >. )
4 rdgeq2 6672 . . . 4  |-  ( <. M ,  ( F `  M ) >.  =  <. N ,  ( F `  N ) >.  ->  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  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. N , 
( F `  N
) >. ) )
53, 4syl 16 . . 3  |-  ( M  =  N  ->  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  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. N , 
( F `  N
) >. ) )
65imaeq1d 5204 . 2  |-  ( M  =  N  ->  ( 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  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. N , 
( F `  N
) >. ) " om ) )
7 df-seq 11326 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
8 df-seq 11326 . 2  |-  seq  N
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. N , 
( F `  N
) >. ) " om )
96, 7, 83eqtr4g 2495 1  |-  ( M  =  N  ->  seq  M (  .+  ,  F
)  =  seq  N
(  .+  ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   _Vcvv 2958   <.cop 3819   omcom 4847   "cima 4883   ` cfv 5456  (class class class)co 6083    e. cmpt2 6085   reccrdg 6669   1c1 8993    + caddc 8995    seq cseq 11325
This theorem is referenced by:  seqeq1d  11331  seqfn  11337  seq1  11338  seqp1  11340  seqf1olem2  11365  seqid  11370  seqz  11373  iserex  12452  summolem2  12512  summo  12513  zsum  12514  isumsplit  12622  ege2le3  12694  gsumval2a  14784  leibpi  20784  ntrivcvg  25227  ntrivcvgn0  25228  ntrivcvgtail  25230  ntrivcvgmullem  25231  prodmolem2  25263  prodmo  25264  zprod  25265  fprodntriv  25270  stirlinglem12  27812
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-cnv 4888  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fv 5464  df-recs 6635  df-rdg 6670  df-seq 11326
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