MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  seqval Structured version   Unicode version

Theorem seqval 11822
Description: Value of the sequence builder function. (Contributed by Mario Carneiro, 24-Jun-2013.)
Hypothesis
Ref Expression
seqval.1  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
Assertion
Ref Expression
seqval  |-  seq M
(  .+  ,  F
)  =  ran  R
Distinct variable groups:    w, F, x, y, z    w,  .+ , x, y, z    x, M, y
Allowed substitution hints:    R( x, y, z, w)    M( z, w)

Proof of Theorem seqval
StepHypRef Expression
1 df-ima 4858 . 2  |-  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )  =  ran  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )  |`  om )
2 df-seq 11812 . 2  |-  seq M
(  .+  ,  F
)  =  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) " om )
3 seqval.1 . . . 4  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
4 eqid 2443 . . . . . . 7  |-  _V  =  _V
5 vex 2980 . . . . . . . . 9  |-  x  e. 
_V
6 vex 2980 . . . . . . . . 9  |-  y  e. 
_V
7 oveq1 6103 . . . . . . . . . . . 12  |-  ( z  =  x  ->  (
z  +  1 )  =  ( x  + 
1 ) )
87fveq2d 5700 . . . . . . . . . . 11  |-  ( z  =  x  ->  ( F `  ( z  +  1 ) )  =  ( F `  ( x  +  1
) ) )
98oveq2d 6112 . . . . . . . . . 10  |-  ( z  =  x  ->  (
w  .+  ( F `  ( z  +  1 ) ) )  =  ( w  .+  ( F `  ( x  +  1 ) ) ) )
10 oveq1 6103 . . . . . . . . . 10  |-  ( w  =  y  ->  (
w  .+  ( F `  ( x  +  1 ) ) )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
11 eqid 2443 . . . . . . . . . 10  |-  ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) )  =  ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) )
12 ovex 6121 . . . . . . . . . 10  |-  ( y 
.+  ( F `  ( x  +  1
) ) )  e. 
_V
139, 10, 11, 12ovmpt2 6231 . . . . . . . . 9  |-  ( ( x  e.  _V  /\  y  e.  _V )  ->  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) ) )
145, 6, 13mp2an 672 . . . . . . . 8  |-  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y )  =  ( y  .+  ( F `  ( x  +  1 ) ) )
1514opeq2i 4068 . . . . . . 7  |-  <. (
x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) ) y ) >.  =  <. ( x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >.
164, 4, 15mpt2eq123i 6154 . . . . . 6  |-  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w 
.+  ( F `  ( z  +  1 ) ) ) ) y ) >. )  =  ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. )
17 rdgeq1 6872 . . . . . 6  |-  ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
)  =  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( y  .+  ( F `  ( x  +  1 ) ) ) >. )  ->  rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x ( z  e. 
_V ,  w  e. 
_V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )
>. ) ,  <. M , 
( F `  M
) >. )  =  rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. ) )
1816, 17ax-mp 5 . . . . 5  |-  rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x ( z  e.  _V ,  w  e.  _V  |->  ( w  .+  ( F `
 ( z  +  1 ) ) ) ) y ) >.
) ,  <. M , 
( F `  M
) >. )  =  rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( y  .+  ( F `
 ( x  + 
1 ) ) )
>. ) ,  <. M , 
( F `  M
) >. )
1918reseq1i 5111 . . . 4  |-  ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x ( z  e. 
_V ,  w  e. 
_V  |->  ( w  .+  ( F `  ( z  +  1 ) ) ) ) y )
>. ) ,  <. M , 
( F `  M
) >. )  |`  om )  =  ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
203, 19eqtri 2463 . . 3  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
2120rneqi 5071 . 2  |-  ran  R  =  ran  ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( y 
.+  ( F `  ( x  +  1
) ) ) >.
) ,  <. M , 
( F `  M
) >. )  |`  om )
221, 2, 213eqtr4i 2473 1  |-  seq M
(  .+  ,  F
)  =  ran  R
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369    e. wcel 1756   _Vcvv 2977   <.cop 3888   ran crn 4846    |` cres 4847   "cima 4848   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   omcom 6481   reccrdg 6870   1c1 9288    + caddc 9290    seqcseq 11811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-recs 6837  df-rdg 6871  df-seq 11812
This theorem is referenced by:  seqfn  11823  seq1  11824  seqp1  11826
  Copyright terms: Public domain W3C validator