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Theorem seqeq1 11065
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 5541 . . . . 5  |-  ( M  =  N  ->  ( F `  M )  =  ( F `  N ) )
2 opeq12 3814 . . . . 5  |-  ( ( M  =  N  /\  ( F `  M )  =  ( F `  N ) )  ->  <. M ,  ( F `
 M ) >.  =  <. N ,  ( F `  N )
>. )
31, 2mpdan 649 . . . 4  |-  ( M  =  N  ->  <. M , 
( F `  M
) >.  =  <. N , 
( F `  N
) >. )
4 rdgeq2 6441 . . . 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 15 . . 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 5027 . 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 11063 . 2  |-  seq  M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
8 df-seq 11063 . 2  |-  seq  N
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. N , 
( F `  N
) >. ) " om )
96, 7, 83eqtr4g 2353 1  |-  ( M  =  N  ->  seq  M (  .+  ,  F
)  =  seq  N
(  .+  ,  F
) )
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:  seqeq1d  11068  seqfn  11074  seq1  11075  seqp1  11077  seqf1olem2  11102  seqid  11107  seqz  11110  iserex  12146  summolem2  12205  summo  12206  zsum  12207  isumsplit  12315  ege2le3  12387  gsumval2a  14475  leibpi  20254  prodmolem2  24158  prodmo  24159  zprod  24160  zprodn0  24162  dffprod  25422  stirlinglem12  27937
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-recs 6404  df-rdg 6439  df-seq 11063
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