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Theorem iseqeq1 9846
Description: Equality theorem for the sequence builder operation.

New proofs should use seqeq1 9849 instead (together with iseqsst 9874 or iseqseq3 9890 if need be).

(Contributed by Jim Kingdon, 30-May-2020.) (New usage is discouraged.)

Assertion
Ref Expression
iseqeq1  |-  ( M  =  N  ->  seq M (  .+  ,  F ,  S )  =  seq N (  .+  ,  F ,  S ) )

Proof of Theorem iseqeq1
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . . 6  |-  ( M  =  N  ->  M  =  N )
2 fveq2 5299 . . . . . 6  |-  ( M  =  N  ->  ( F `  M )  =  ( F `  N ) )
31, 2opeq12d 3628 . . . . 5  |-  ( M  =  N  ->  <. M , 
( F `  M
) >.  =  <. N , 
( F `  N
) >. )
4 freceq2 6150 . . . . 5  |-  ( <. M ,  ( F `  M ) >.  =  <. N ,  ( F `  N ) >.  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
53, 4syl 14 . . . 4  |-  ( M  =  N  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
6 fveq2 5299 . . . . . 6  |-  ( M  =  N  ->  ( ZZ>=
`  M )  =  ( ZZ>= `  N )
)
7 eqid 2088 . . . . . 6  |-  S  =  S
8 mpt2eq12 5701 . . . . . 6  |-  ( ( ( ZZ>= `  M )  =  ( ZZ>= `  N
)  /\  S  =  S )  ->  (
x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  N ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) )
96, 7, 8sylancl 404 . . . . 5  |-  ( M  =  N  ->  (
x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  N ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) )
10 freceq1 6149 . . . . 5  |-  ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  N ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. N ,  ( F `  N ) >. )  = frec ( ( x  e.  ( ZZ>= `  N ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
119, 10syl 14 . . . 4  |-  ( M  =  N  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. N ,  ( F `  N ) >. )  = frec ( ( x  e.  ( ZZ>= `  N ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
125, 11eqtrd 2120 . . 3  |-  ( M  =  N  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  N ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
1312rneqd 4660 . 2  |-  ( M  =  N  ->  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  =  ran frec ( ( x  e.  (
ZZ>= `  N ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. ) )
14 df-iseq 9841 . 2  |-  seq M
(  .+  ,  F ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
15 df-iseq 9841 . 2  |-  seq N
(  .+  ,  F ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  N ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. N , 
( F `  N
) >. )
1613, 14, 153eqtr4g 2145 1  |-  ( M  =  N  ->  seq M (  .+  ,  F ,  S )  =  seq N (  .+  ,  F ,  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   <.cop 3447   ran crn 4437   ` cfv 5010  (class class class)co 5644    |-> cmpt2 5646  freccfrec 6147   1c1 7341    + caddc 7343   ZZ>=cuz 9009    seqcseq4 9839
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-br 3844  df-opab 3898  df-mpt 3899  df-cnv 4444  df-dm 4446  df-rn 4447  df-res 4448  df-iota 4975  df-fv 5018  df-oprab 5648  df-mpt2 5649  df-recs 6062  df-frec 6148  df-iseq 9841
This theorem is referenced by:  seqeq1  9849  iseqid  9927  iseqz  9931  ibcval5  10159  bcn2  10160  isummolem2  10759  isummo  10760  zisum  10761
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