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Theorem om2uzrdg 11019
Description: A helper lemma for the value of a recursive definition generator on upper integers (typically either  NN or  NN0) with characteristic function  F ( x ,  y ) and initial value  A. Normally  F is a function on the partition, and  A is a member of the partition. See also comment in om2uz0i 11010. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)
Hypotheses
Ref Expression
om2uz.1  |-  C  e.  ZZ
om2uz.2  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
uzrdg.1  |-  A  e. 
_V
uzrdg.2  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om )
Assertion
Ref Expression
om2uzrdg  |-  ( B  e.  om  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `
 B ) )
>. )
Distinct variable groups:    y, A    x, y, C    y, G    x, F, y
Allowed substitution hints:    A( x)    B( x, y)    R( x, y)    G( x)

Proof of Theorem om2uzrdg
Dummy variables  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . 3  |-  ( z  =  (/)  ->  ( R `
 z )  =  ( R `  (/) ) )
2 fveq2 5525 . . . 4  |-  ( z  =  (/)  ->  ( G `
 z )  =  ( G `  (/) ) )
31fveq2d 5529 . . . 4  |-  ( z  =  (/)  ->  ( 2nd `  ( R `  z
) )  =  ( 2nd `  ( R `
 (/) ) ) )
42, 3opeq12d 3804 . . 3  |-  ( z  =  (/)  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. )
51, 4eqeq12d 2297 . 2  |-  ( z  =  (/)  ->  ( ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `
 z ) )
>. 
<->  ( R `  (/) )  = 
<. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>. ) )
6 fveq2 5525 . . 3  |-  ( z  =  v  ->  ( R `  z )  =  ( R `  v ) )
7 fveq2 5525 . . . 4  |-  ( z  =  v  ->  ( G `  z )  =  ( G `  v ) )
86fveq2d 5529 . . . 4  |-  ( z  =  v  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  v )
) )
97, 8opeq12d 3804 . . 3  |-  ( z  =  v  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. )
106, 9eqeq12d 2297 . 2  |-  ( z  =  v  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  v
)  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >. ) )
11 fveq2 5525 . . 3  |-  ( z  =  suc  v  -> 
( R `  z
)  =  ( R `
 suc  v )
)
12 fveq2 5525 . . . 4  |-  ( z  =  suc  v  -> 
( G `  z
)  =  ( G `
 suc  v )
)
1311fveq2d 5529 . . . 4  |-  ( z  =  suc  v  -> 
( 2nd `  ( R `  z )
)  =  ( 2nd `  ( R `  suc  v ) ) )
1412, 13opeq12d 3804 . . 3  |-  ( z  =  suc  v  ->  <. ( G `  z
) ,  ( 2nd `  ( R `  z
) ) >.  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
)
1511, 14eqeq12d 2297 . 2  |-  ( z  =  suc  v  -> 
( ( R `  z )  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  suc  v )  =  <. ( G `  suc  v
) ,  ( 2nd `  ( R `  suc  v ) ) >.
) )
16 fveq2 5525 . . 3  |-  ( z  =  B  ->  ( R `  z )  =  ( R `  B ) )
17 fveq2 5525 . . . 4  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
1816fveq2d 5529 . . . 4  |-  ( z  =  B  ->  ( 2nd `  ( R `  z ) )  =  ( 2nd `  ( R `  B )
) )
1917, 18opeq12d 3804 . . 3  |-  ( z  =  B  ->  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >.  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. )
2016, 19eqeq12d 2297 . 2  |-  ( z  =  B  ->  (
( R `  z
)  =  <. ( G `  z ) ,  ( 2nd `  ( R `  z )
) >. 
<->  ( R `  B
)  =  <. ( G `  B ) ,  ( 2nd `  ( R `  B )
) >. ) )
21 uzrdg.2 . . . . 5  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om )
2221fveq1i 5526 . . . 4  |-  ( R `
 (/) )  =  ( ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  (/) )
23 opex 4237 . . . . 5  |-  <. C ,  A >.  e.  _V
24 fr0g 6448 . . . . 5  |-  ( <. C ,  A >.  e. 
_V  ->  ( ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. )  |`  om ) `  (/) )  =  <. C ,  A >. )
2523, 24ax-mp 8 . . . 4  |-  ( ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om ) `  (/) )  =  <. C ,  A >.
2622, 25eqtri 2303 . . 3  |-  ( R `
 (/) )  =  <. C ,  A >.
27 om2uz.1 . . . . 5  |-  C  e.  ZZ
28 om2uz.2 . . . . 5  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
2927, 28om2uz0i 11010 . . . 4  |-  ( G `
 (/) )  =  C
3026fveq2i 5528 . . . . 5  |-  ( 2nd `  ( R `  (/) ) )  =  ( 2nd `  <. C ,  A >. )
3127elexi 2797 . . . . . 6  |-  C  e. 
_V
32 uzrdg.1 . . . . . 6  |-  A  e. 
_V
3331, 32op2nd 6129 . . . . 5  |-  ( 2nd `  <. C ,  A >. )  =  A
3430, 33eqtri 2303 . . . 4  |-  ( 2nd `  ( R `  (/) ) )  =  A
3529, 34opeq12i 3801 . . 3  |-  <. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>.  =  <. C ,  A >.
3626, 35eqtr4i 2306 . 2  |-  ( R `
 (/) )  =  <. ( G `  (/) ) ,  ( 2nd `  ( R `  (/) ) )
>.
37 frsuc 6449 . . . . . 6  |-  ( v  e.  om  ->  (
( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  suc  v )  =  ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  ( ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om ) `  v ) ) )
3821fveq1i 5526 . . . . . 6  |-  ( R `
 suc  v )  =  ( ( rec ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ,  <. C ,  A >. )  |`  om ) `  suc  v )
3921fveq1i 5526 . . . . . . 7  |-  ( R `
 v )  =  ( ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  v )
4039fveq2i 5528 . . . . . 6  |-  ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  v ) )  =  ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  ( ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om ) `  v ) )
4137, 38, 403eqtr4g 2340 . . . . 5  |-  ( v  e.  om  ->  ( R `  suc  v )  =  ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  v
) ) )
42 fveq2 5525 . . . . . 6  |-  ( ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>.  ->  ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  v
) )  =  ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 v ) ,  ( 2nd `  ( R `  v )
) >. ) )
43 df-ov 5861 . . . . . . 7  |-  ( ( G `  v ) ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  v )
) )  =  ( ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 v ) ,  ( 2nd `  ( R `  v )
) >. )
44 fvex 5539 . . . . . . . 8  |-  ( G `
 v )  e. 
_V
45 fvex 5539 . . . . . . . 8  |-  ( 2nd `  ( R `  v
) )  e.  _V
46 oveq1 5865 . . . . . . . . . 10  |-  ( w  =  ( G `  v )  ->  (
w  +  1 )  =  ( ( G `
 v )  +  1 ) )
47 oveq1 5865 . . . . . . . . . 10  |-  ( w  =  ( G `  v )  ->  (
w F z )  =  ( ( G `
 v ) F z ) )
4846, 47opeq12d 3804 . . . . . . . . 9  |-  ( w  =  ( G `  v )  ->  <. (
w  +  1 ) ,  ( w F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F z ) >. )
49 oveq2 5866 . . . . . . . . . 10  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  ( ( G `  v ) F z )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
5049opeq2d 3803 . . . . . . . . 9  |-  ( z  =  ( 2nd `  ( R `  v )
)  ->  <. ( ( G `  v )  +  1 ) ,  ( ( G `  v ) F z ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
51 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x  +  1 )  =  ( w  + 
1 ) )
52 oveq1 5865 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
x F y )  =  ( w F y ) )
5351, 52opeq12d 3804 . . . . . . . . . 10  |-  ( x  =  w  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( w  +  1 ) ,  ( w F y ) >. )
54 oveq2 5866 . . . . . . . . . . 11  |-  ( y  =  z  ->  (
w F y )  =  ( w F z ) )
5554opeq2d 3803 . . . . . . . . . 10  |-  ( y  =  z  ->  <. (
w  +  1 ) ,  ( w F y ) >.  =  <. ( w  +  1 ) ,  ( w F z ) >. )
5653, 55cbvmpt2v 5926 . . . . . . . . 9  |-  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. )  =  ( w  e. 
_V ,  z  e. 
_V  |->  <. ( w  + 
1 ) ,  ( w F z )
>. )
57 opex 4237 . . . . . . . . 9  |-  <. (
( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >.  e.  _V
5848, 50, 56, 57ovmpt2 5983 . . . . . . . 8  |-  ( ( ( G `  v
)  e.  _V  /\  ( 2nd `  ( R `
 v ) )  e.  _V )  -> 
( ( G `  v ) ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. )
( 2nd `  ( R `  v )
) )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
5944, 45, 58mp2an 653 . . . . . . 7  |-  ( ( G `  v ) ( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  v )
) )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >.
6043, 59eqtr3i 2305 . . . . . 6  |-  ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  <. ( G `
 v ) ,  ( 2nd `  ( R `  v )
) >. )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >.
6142, 60syl6eq 2331 . . . . 5  |-  ( ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>.  ->  ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  ( R `  v
) )  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
6241, 61sylan9eq 2335 . . . 4  |-  ( ( v  e.  om  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  <. ( ( G `
 v )  +  1 ) ,  ( ( G `  v
) F ( 2nd `  ( R `  v
) ) ) >.
)
6327, 28om2uzsuci 11011 . . . . . 6  |-  ( v  e.  om  ->  ( G `  suc  v )  =  ( ( G `
 v )  +  1 ) )
6463adantr 451 . . . . 5  |-  ( ( v  e.  om  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( G `
 suc  v )  =  ( ( G `
 v )  +  1 ) )
6562fveq2d 5529 . . . . . 6  |-  ( ( v  e.  om  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( 2nd `  ( R `  suc  v ) )  =  ( 2nd `  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
)
66 ovex 5883 . . . . . . 7  |-  ( ( G `  v )  +  1 )  e. 
_V
67 ovex 5883 . . . . . . 7  |-  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )  e.  _V
6866, 67op2nd 6129 . . . . . 6  |-  ( 2nd `  <. ( ( G `
 v )  +  1 ) ,  ( ( G `  v
) F ( 2nd `  ( R `  v
) ) ) >.
)  =  ( ( G `  v ) F ( 2nd `  ( R `  v )
) )
6965, 68syl6eq 2331 . . . . 5  |-  ( ( v  e.  om  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( 2nd `  ( R `  suc  v ) )  =  ( ( G `  v ) F ( 2nd `  ( R `
 v ) ) ) )
7064, 69opeq12d 3804 . . . 4  |-  ( ( v  e.  om  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  <. ( G `  suc  v ) ,  ( 2nd `  ( R `  suc  v ) ) >.  =  <. ( ( G `  v
)  +  1 ) ,  ( ( G `
 v ) F ( 2nd `  ( R `  v )
) ) >. )
7162, 70eqtr4d 2318 . . 3  |-  ( ( v  e.  om  /\  ( R `  v )  =  <. ( G `  v ) ,  ( 2nd `  ( R `
 v ) )
>. )  ->  ( R `
 suc  v )  =  <. ( G `  suc  v ) ,  ( 2nd `  ( R `
 suc  v )
) >. )
7271ex 423 . 2  |-  ( v  e.  om  ->  (
( R `  v
)  =  <. ( G `  v ) ,  ( 2nd `  ( R `  v )
) >.  ->  ( R `  suc  v )  = 
<. ( G `  suc  v ) ,  ( 2nd `  ( R `
 suc  v )
) >. ) )
735, 10, 15, 20, 36, 72finds 4682 1  |-  ( B  e.  om  ->  ( R `  B )  =  <. ( G `  B ) ,  ( 2nd `  ( R `
 B ) )
>. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   <.cop 3643    e. cmpt 4077   suc csuc 4394   omcom 4656    |` cres 4691   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   2ndc2nd 6121   reccrdg 6422   1c1 8738    + caddc 8740   ZZcz 10024
This theorem is referenced by:  uzrdglem  11020  uzrdgfni  11021  uzrdgsuci  11023
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-recs 6388  df-rdg 6423
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