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Theorem seqeq3 10253
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 simp1 982 . . . . . . . 8  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  _V )  ->  F  =  G )
21fveq1d 5430 . . . . . . 7  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  _V )  ->  ( F `  ( x  +  1 ) )  =  ( G `  ( x  +  1
) ) )
32oveq2d 5797 . . . . . 6  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  _V )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( G `  ( x  +  1 ) ) ) )
43opeq2d 3719 . . . . 5  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  _V )  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. )
54mpoeq3dva 5842 . . . 4  |-  ( F  =  G  ->  (
x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. ) )
6 fveq1 5427 . . . . 5  |-  ( F  =  G  ->  ( F `  M )  =  ( G `  M ) )
76opeq2d 3719 . . . 4  |-  ( F  =  G  ->  <. M , 
( F `  M
) >.  =  <. M , 
( G `  M
) >. )
8 freceq1 6296 . . . . 5  |-  ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. )  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
9 freceq2 6297 . . . . 5  |-  ( <. M ,  ( F `  M ) >.  =  <. M ,  ( G `  M ) >.  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
108, 9sylan9eq 2193 . . . 4  |-  ( ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
)  =  ( x  e.  ( ZZ>= `  M
) ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( G `
 ( x  + 
1 ) ) )
>. )  /\  <. M , 
( F `  M
) >.  =  <. M , 
( G `  M
) >. )  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
115, 7, 10syl2anc 409 . . 3  |-  ( F  =  G  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
1211rneqd 4775 . 2  |-  ( F  =  G  ->  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
13 df-seqfrec 10249 . 2  |-  seq M
(  .+  ,  F
)  =  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
14 df-seqfrec 10249 . 2  |-  seq M
(  .+  ,  G
)  =  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. )
1512, 13, 143eqtr4g 2198 1  |-  ( F  =  G  ->  seq M (  .+  ,  F )  =  seq M (  .+  ,  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 963    = wceq 1332    e. wcel 1481   _Vcvv 2689   <.cop 3534   ran crn 4547   ` cfv 5130  (class class class)co 5781    e. cmpo 5783  freccfrec 6294   1c1 7644    + caddc 7646   ZZ>=cuz 9349    seqcseq 10248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-cnv 4554  df-dm 4556  df-rn 4557  df-res 4558  df-iota 5095  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-recs 6209  df-frec 6295  df-seqfrec 10249
This theorem is referenced by:  seqeq3d  10256  cbvsum  11160  fsumadd  11206  cbvprod  11358
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