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Theorem iseqeq2 9922
Description: Equality theorem for the sequence builder operation.

New proofs should use seqeq2 9925 instead (together with iseqsst 9949 or iseqseq3 9965 if need be).

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

Assertion
Ref Expression
iseqeq2  |-  (  .+  =  Q  ->  seq M
(  .+  ,  F ,  S )  =  seq M ( Q ,  F ,  S )
)

Proof of Theorem iseqeq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 944 . . . . . . 7  |-  ( ( 
.+  =  Q  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  S )  ->  .+  =  Q )
21oveqd 5685 . . . . . 6  |-  ( ( 
.+  =  Q  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  S )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  =  ( y Q ( F `  ( x  +  1 ) ) ) )
32opeq2d 3637 . . . . 5  |-  ( ( 
.+  =  Q  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  S )  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
43mpt2eq3dva 5729 . . . 4  |-  (  .+  =  Q  ->  ( x  e.  ( ZZ>= `  M
) ,  y  e.  S  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )  =  (
x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
)
5 freceq1 6173 . . . 4  |-  ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  S  |->  <. (
x  +  1 ) ,  ( y Q ( 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.  S  |->  <.
( x  +  1 ) ,  ( y Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
64, 5syl 14 . . 3  |-  (  .+  =  Q  -> 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 Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
76rneqd 4679 . 2  |-  (  .+  =  Q  ->  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 Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
8 df-iseq 9916 . 2  |-  seq M
(  .+  ,  F ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
9 df-iseq 9916 . 2  |-  seq M
( Q ,  F ,  S )  =  ran frec ( ( x  e.  (
ZZ>= `  M ) ,  y  e.  S  |->  <.
( x  +  1 ) ,  ( y Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
107, 8, 93eqtr4g 2146 1  |-  (  .+  =  Q  ->  seq M
(  .+  ,  F ,  S )  =  seq M ( Q ,  F ,  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 925    = wceq 1290    e. wcel 1439   <.cop 3455   ran crn 4455   ` cfv 5030  (class class class)co 5668    |-> cmpt2 5670  freccfrec 6171   1c1 7414    + caddc 7416   ZZ>=cuz 9082    seqcseq4 9914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-mpt 3909  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-iota 4995  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-recs 6086  df-frec 6172  df-iseq 9916
This theorem is referenced by:  seqeq2  9925
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