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Theorem iseqeq4 9785
Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
iseqeq4  |-  ( S  =  T  ->  seq M (  .+  ,  F ,  S )  =  seq M (  .+  ,  F ,  T ) )

Proof of Theorem iseqeq4
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2085 . . . . 5  |-  ( ZZ>= `  M )  =  (
ZZ>= `  M )
2 mpt2eq12 5666 . . . . 5  |-  ( ( ( ZZ>= `  M )  =  ( ZZ>= `  M
)  /\  S  =  T )  ->  (
x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) )
31, 2mpan 415 . . . 4  |-  ( S  =  T  ->  (
x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) )
4 freceq1 6111 . . . 4  |-  ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  T  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
53, 4syl 14 . . 3  |-  ( S  =  T  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  T  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
65rneqd 4632 . 2  |-  ( S  =  T  ->  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
7 df-iseq 9780 . 2  |-  seq M
(  .+  ,  F ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
8 df-iseq 9780 . 2  |-  seq M
(  .+  ,  F ,  T )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  T  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
96, 7, 83eqtr4g 2142 1  |-  ( S  =  T  ->  seq M (  .+  ,  F ,  S )  =  seq M (  .+  ,  F ,  T ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   <.cop 3434   ran crn 4412   ` cfv 4981  (class class class)co 5613    |-> cmpt2 5615  freccfrec 6109   1c1 7295    + caddc 7297   ZZ>=cuz 8951    seqcseq 9779
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-mpt 3876  df-cnv 4419  df-dm 4421  df-rn 4422  df-res 4423  df-iota 4946  df-fv 4989  df-oprab 5617  df-mpt2 5618  df-recs 6024  df-frec 6110  df-iseq 9780
This theorem is referenced by: (None)
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