ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  iseqeq3 Unicode version

Theorem iseqeq3 9759
Description: Equality theorem for the sequence builder operation. (Contributed by Jim Kingdon, 30-May-2020.)
Assertion
Ref Expression
iseqeq3  |-  ( F  =  G  ->  seq M (  .+  ,  F ,  S )  =  seq M (  .+  ,  G ,  S ) )

Proof of Theorem iseqeq3
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 941 . . . . . . . 8  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  S )  ->  F  =  G )
21fveq1d 5258 . . . . . . 7  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  S )  ->  ( F `  ( x  +  1 ) )  =  ( G `  ( x  +  1
) ) )
32oveq2d 5610 . . . . . 6  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  S )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( G `  ( x  +  1 ) ) ) )
43opeq2d 3606 . . . . 5  |-  ( ( F  =  G  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  S )  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. )
54mpt2eq3dva 5651 . . . 4  |-  ( F  =  G  ->  (
x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. ) )
6 fveq1 5255 . . . . 5  |-  ( F  =  G  ->  ( F `  M )  =  ( G `  M ) )
76opeq2d 3606 . . . 4  |-  ( F  =  G  ->  <. M , 
( F `  M
) >.  =  <. M , 
( G `  M
) >. )
8 freceq1 6092 . . . . 5  |-  ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( 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.  S  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
9 freceq2 6093 . . . . 5  |-  ( <. M ,  ( F `  M ) >.  =  <. M ,  ( G `  M ) >.  -> frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( G `  ( x  +  1 ) ) ) >. ) ,  <. M ,  ( F `  M ) >. )  = frec ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |-> 
<. ( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
108, 9sylan9eq 2137 . . . 4  |-  ( ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
)  =  ( x  e.  ( ZZ>= `  M
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( y  .+  ( G `
 ( x  + 
1 ) ) )
>. )  /\  <. M , 
( F `  M
) >.  =  <. M , 
( G `  M
) >. )  -> 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 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
115, 7, 10syl2anc 403 . . 3  |-  ( F  =  G  -> 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 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
1211rneqd 4625 . 2  |-  ( F  =  G  ->  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.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. ) )
13 df-iseq 9755 . 2  |-  seq M
(  .+  ,  F ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
14 df-iseq 9755 . 2  |-  seq M
(  .+  ,  G ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( G `  ( x  +  1
) ) ) >.
) ,  <. M , 
( G `  M
) >. )
1512, 13, 143eqtr4g 2142 1  |-  ( F  =  G  ->  seq M (  .+  ,  F ,  S )  =  seq M (  .+  ,  G ,  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 922    = wceq 1287    e. wcel 1436   <.cop 3428   ran crn 4405   ` cfv 4972  (class class class)co 5594    |-> cmpt2 5596  freccfrec 6090   1c1 7272    + caddc 7274   ZZ>=cuz 8928    seqcseq 9754
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 2616  df-un 2990  df-in 2992  df-ss 2999  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-br 3815  df-opab 3869  df-mpt 3870  df-cnv 4412  df-dm 4414  df-rn 4415  df-res 4416  df-iota 4937  df-fv 4980  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-recs 6005  df-frec 6091  df-iseq 9755
This theorem is referenced by:  expival  9808  sumeq1  10580  sumeq2d  10584  sumeq2  10585
  Copyright terms: Public domain W3C validator