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Theorem seqeq1 11281
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 5687 . . . . 5  |-  ( M  =  N  ->  ( F `  M )  =  ( F `  N ) )
2 opeq12 3946 . . . . 5  |-  ( ( M  =  N  /\  ( F `  M )  =  ( F `  N ) )  ->  <. M ,  ( F `
 M ) >.  =  <. N ,  ( F `  N )
>. )
31, 2mpdan 650 . . . 4  |-  ( M  =  N  ->  <. M , 
( F `  M
) >.  =  <. N , 
( F `  N
) >. )
4 rdgeq2 6629 . . . 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 5161 . 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 11279 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
8 df-seq 11279 . 2  |-  seq  N
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. N , 
( F `  N
) >. ) " om )
96, 7, 83eqtr4g 2461 1  |-  ( M  =  N  ->  seq  M (  .+  ,  F
)  =  seq  N
(  .+  ,  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2916   <.cop 3777   omcom 4804   "cima 4840   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   reccrdg 6626   1c1 8947    + caddc 8949    seq cseq 11278
This theorem is referenced by:  seqeq1d  11284  seqfn  11290  seq1  11291  seqp1  11293  seqf1olem2  11318  seqid  11323  seqz  11326  iserex  12405  summolem2  12465  summo  12466  zsum  12467  isumsplit  12575  ege2le3  12647  gsumval2a  14737  leibpi  20735  ntrivcvg  25178  ntrivcvgn0  25179  ntrivcvgtail  25181  ntrivcvgmullem  25182  prodmolem2  25214  prodmo  25215  zprod  25216  fprodntriv  25221  stirlinglem12  27701
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fv 5421  df-recs 6592  df-rdg 6627  df-seq 11279
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