MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqeq1 Structured version   Unicode version

Theorem seqeq1 12077
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 5865 . . . . 5  |-  ( M  =  N  ->  ( F `  M )  =  ( F `  N ) )
2 opeq12 4215 . . . . 5  |-  ( ( M  =  N  /\  ( F `  M )  =  ( F `  N ) )  ->  <. M ,  ( F `
 M ) >.  =  <. N ,  ( F `  N )
>. )
31, 2mpdan 668 . . . 4  |-  ( M  =  N  ->  <. M , 
( F `  M
) >.  =  <. N , 
( F `  N
) >. )
4 rdgeq2 7078 . . . 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 5335 . 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 12075 . 2  |-  seq M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
8 df-seq 12075 . 2  |-  seq N
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. N , 
( F `  N
) >. ) " om )
96, 7, 83eqtr4g 2533 1  |-  ( M  =  N  ->  seq M (  .+  ,  F )  =  seq N (  .+  ,  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   _Vcvv 3113   <.cop 4033   "cima 5002   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   omcom 6679   reccrdg 7075   1c1 9492    + caddc 9494    seqcseq 12074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fv 5595  df-recs 7042  df-rdg 7076  df-seq 12075
This theorem is referenced by:  seqeq1d  12080  seqfn  12086  seq1  12087  seqp1  12089  seqf1olem2  12114  seqid  12119  seqz  12122  iserex  13441  summolem2  13500  summo  13501  zsum  13502  isumsplit  13614  ege2le3  13686  gsumval2a  15831  leibpi  23017  ntrivcvg  28624  ntrivcvgn0  28625  ntrivcvgtail  28627  ntrivcvgmullem  28628  prodmolem2  28660  prodmo  28661  zprod  28662  fprodntriv  28667  stirlinglem12  31401
  Copyright terms: Public domain W3C validator