ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  seqeq2 Unicode version

Theorem seqeq2 10252
Description: Equality theorem for the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013.)
Assertion
Ref Expression
seqeq2  |-  (  .+  =  Q  ->  seq M
(  .+  ,  F
)  =  seq M
( Q ,  F
) )

Proof of Theorem seqeq2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 982 . . . . . . 7  |-  ( ( 
.+  =  Q  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  _V )  ->  .+  =  Q )
21oveqd 5798 . . . . . 6  |-  ( ( 
.+  =  Q  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  _V )  ->  (
y  .+  ( F `  ( x  +  1 ) ) )  =  ( y Q ( F `  ( x  +  1 ) ) ) )
32opeq2d 3719 . . . . 5  |-  ( ( 
.+  =  Q  /\  x  e.  ( ZZ>= `  M )  /\  y  e.  _V )  ->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.  =  <. ( x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
43mpoeq3dva 5842 . . . 4  |-  (  .+  =  Q  ->  ( x  e.  ( ZZ>= `  M
) ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )  =  (
x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y Q ( F `  (
x  +  1 ) ) ) >. )
)
5 freceq1 6296 . . . 4  |-  ( ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  =  ( x  e.  ( ZZ>= `  M ) ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y Q ( F `  (
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 Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
64, 5syl 14 . . 3  |-  (  .+  =  Q  -> 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 Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
76rneqd 4775 . 2  |-  (  .+  =  Q  ->  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 Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. ) )
8 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
) >. )
9 df-seqfrec 10249 . 2  |-  seq M
( Q ,  F
)  =  ran frec (
( x  e.  (
ZZ>= `  M ) ,  y  e.  _V  |->  <.
( x  +  1 ) ,  ( y Q ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )
107, 8, 93eqtr4g 2198 1  |-  (  .+  =  Q  ->  seq M
(  .+  ,  F
)  =  seq M
( Q ,  F
) )
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:  seqeq2d  10255  resqrex  10829
  Copyright terms: Public domain W3C validator