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Theorem uzrdgsuci 12038
Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12034. (Contributed by Mario Carneiro, 26-Jun-2013.) (Revised by Mario Carneiro, 13-Sep-2013.)
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 )
uzrdg.3  |-  S  =  ran  R
Assertion
Ref Expression
uzrdgsuci  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( B F ( S `
 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)    S( x, y)    G( x)

Proof of Theorem uzrdgsuci
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 om2uz.1 . . . . . 6  |-  C  e.  ZZ
2 om2uz.2 . . . . . 6  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
3 uzrdg.1 . . . . . 6  |-  A  e. 
_V
4 uzrdg.2 . . . . . 6  |-  R  =  ( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om )
5 uzrdg.3 . . . . . 6  |-  S  =  ran  R
61, 2, 3, 4, 5uzrdgfni 12036 . . . . 5  |-  S  Fn  ( ZZ>= `  C )
7 fnfun 5677 . . . . 5  |-  ( S  Fn  ( ZZ>= `  C
)  ->  Fun  S )
86, 7ax-mp 5 . . . 4  |-  Fun  S
9 peano2uz 11133 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
101, 2, 3, 4uzrdglem 12035 . . . . . 6  |-  ( ( B  +  1 )  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  ran  R )
119, 10syl 16 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  ran  R )
1211, 5syl6eleqr 2566 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  S
)
13 funopfv 5906 . . . 4  |-  ( Fun 
S  ->  ( <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  S  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
148, 12, 13mpsyl 63 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 ( `' G `  ( B  +  1 ) ) ) ) )
151, 2om2uzf1oi 12031 . . . . . . . 8  |-  G : om
-1-1-onto-> ( ZZ>= `  C )
16 f1ocnvdm 6175 . . . . . . . 8  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
1715, 16mpan 670 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( `' G `  B )  e.  om )
18 peano2 6699 . . . . . . 7  |-  ( ( `' G `  B )  e.  om  ->  suc  ( `' G `  B )  e.  om )
1917, 18syl 16 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  suc  ( `' G `  B )  e.  om )
201, 2om2uzsuci 12026 . . . . . . . 8  |-  ( ( `' G `  B )  e.  om  ->  ( G `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
2117, 20syl 16 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
22 f1ocnvfv2 6170 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
2315, 22mpan 670 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  ( `' G `  B ) )  =  B )
2423oveq1d 6298 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  =  ( B  + 
1 ) )
2521, 24eqtrd 2508 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 ) )
26 f1ocnvfv 6171 . . . . . . 7  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  suc  ( `' G `  B )  e.  om )  ->  ( ( G `
 suc  ( `' G `  B )
)  =  ( B  +  1 )  -> 
( `' G `  ( B  +  1
) )  =  suc  ( `' G `  B ) ) )
2715, 26mpan 670 . . . . . 6  |-  ( suc  ( `' G `  B )  e.  om  ->  ( ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 )  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) ) )
2819, 25, 27sylc 60 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) )
2928fveq2d 5869 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( R `  ( `' G `  ( B  +  1
) ) )  =  ( R `  suc  ( `' G `  B ) ) )
3029fveq2d 5869 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
3114, 30eqtrd 2508 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
32 frsuc 7102 . . . . . . . 8  |-  ( ( `' G `  B )  e.  om  ->  (
( rec ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  suc  ( `' G `  B ) )  =  ( ( 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 ) `  ( `' G `  B ) ) ) )
334fveq1i 5866 . . . . . . . 8  |-  ( R `
 suc  ( `' G `  B )
)  =  ( ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om ) `  suc  ( `' G `  B ) )
344fveq1i 5866 . . . . . . . . 9  |-  ( R `
 ( `' G `  B ) )  =  ( ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  ( `' G `  B ) )
3534fveq2i 5868 . . . . . . . 8  |-  ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  ( `' G `  B ) ) )  =  ( ( 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 ) `  ( `' G `  B ) ) )
3632, 33, 353eqtr4g 2533 . . . . . . 7  |-  ( ( `' G `  B )  e.  om  ->  ( R `  suc  ( `' G `  B ) )  =  ( ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  ( `' G `  B ) ) ) )
371, 2, 3, 4om2uzrdg 12034 . . . . . . . . 9  |-  ( ( `' G `  B )  e.  om  ->  ( R `  ( `' G `  B )
)  =  <. ( G `  ( `' G `  B )
) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
3837fveq2d 5869 . . . . . . . 8  |-  ( ( `' G `  B )  e.  om  ->  (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  ( `' G `  B ) ) )  =  ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) `  <. ( G `  ( `' G `  B ) ) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
)
39 df-ov 6286 . . . . . . . 8  |-  ( ( G `  ( `' G `  B ) ) ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  ( R `  ( `' G `  B )
) ) )  =  ( ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) `  <. ( G `  ( `' G `  B )
) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
4038, 39syl6eqr 2526 . . . . . . 7  |-  ( ( `' G `  B )  e.  om  ->  (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) `  ( R `  ( `' G `  B ) ) )  =  ( ( G `
 ( `' G `  B ) ) ( x  e.  _V , 
y  e.  _V  |->  <.
( x  +  1 ) ,  ( x F y ) >.
) ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
4136, 40eqtrd 2508 . . . . . 6  |-  ( ( `' G `  B )  e.  om  ->  ( R `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) ) ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
42 fvex 5875 . . . . . . 7  |-  ( G `
 ( `' G `  B ) )  e. 
_V
43 fvex 5875 . . . . . . 7  |-  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  _V
44 oveq1 6290 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z  +  1 )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
45 oveq1 6290 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z F w )  =  ( ( G `  ( `' G `  B ) ) F w ) )
4644, 45opeq12d 4221 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  <. ( z  +  1 ) ,  ( z F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >. )
47 oveq2 6291 . . . . . . . . 9  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  (
( G `  ( `' G `  B ) ) F w )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
4847opeq2d 4220 . . . . . . . 8  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
49 oveq1 6290 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  +  1 )  =  ( z  +  1 ) )
50 oveq1 6290 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x F y )  =  ( z F y ) )
5149, 50opeq12d 4221 . . . . . . . . 9  |-  ( x  =  z  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( z  +  1 ) ,  ( z F y ) >. )
52 oveq2 6291 . . . . . . . . . 10  |-  ( y  =  w  ->  (
z F y )  =  ( z F w ) )
5352opeq2d 4220 . . . . . . . . 9  |-  ( y  =  w  ->  <. (
z  +  1 ) ,  ( z F y ) >.  =  <. ( z  +  1 ) ,  ( z F w ) >. )
5451, 53cbvmpt2v 6360 . . . . . . . 8  |-  ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. )  =  ( z  e. 
_V ,  w  e. 
_V  |->  <. ( z  +  1 ) ,  ( z F w )
>. )
55 opex 4711 . . . . . . . 8  |-  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  _V
5646, 48, 54, 55ovmpt2 6421 . . . . . . 7  |-  ( ( ( G `  ( `' G `  B ) )  e.  _V  /\  ( 2nd `  ( R `
 ( `' G `  B ) ) )  e.  _V )  -> 
( ( G `  ( `' G `  B ) ) ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  ( R `  ( `' G `  B )
) ) )  = 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
5742, 43, 56mp2an 672 . . . . . 6  |-  ( ( G `  ( `' G `  B ) ) ( x  e. 
_V ,  y  e. 
_V  |->  <. ( x  + 
1 ) ,  ( x F y )
>. ) ( 2nd `  ( R `  ( `' G `  B )
) ) )  = 
<. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
5841, 57syl6eq 2524 . . . . 5  |-  ( ( `' G `  B )  e.  om  ->  ( R `  suc  ( `' G `  B ) )  =  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
5958fveq2d 5869 . . . 4  |-  ( ( `' G `  B )  e.  om  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( 2nd `  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
) )
60 ovex 6308 . . . . 5  |-  ( ( G `  ( `' G `  B ) )  +  1 )  e.  _V
61 ovex 6308 . . . . 5  |-  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e. 
_V
6260, 61op2nd 6793 . . . 4  |-  ( 2nd `  <. ( ( G `
 ( `' G `  B ) )  +  1 ) ,  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) >.
)  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )
6359, 62syl6eq 2524 . . 3  |-  ( ( `' G `  B )  e.  om  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
6417, 63syl 16 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
651, 2, 3, 4uzrdglem 12035 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  ran  R )
6665, 5syl6eleqr 2566 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  S
)
67 funopfv 5906 . . . . 5  |-  ( Fun 
S  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  S  ->  ( S `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
688, 66, 67mpsyl 63 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  B )  =  ( 2nd `  ( R `
 ( `' G `  B ) ) ) )
6968eqcomd 2475 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  =  ( S `  B ) )
7023, 69oveq12d 6301 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  =  ( B F ( S `  B ) ) )
7131, 64, 703eqtrd 2512 1  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( B F ( S `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033    |-> cmpt 4505   suc csuc 4880   `'ccnv 4998   ran crn 5000    |` cres 5001   Fun wfun 5581    Fn wfn 5582   -1-1-onto->wf1o 5586   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   omcom 6679   2ndc2nd 6783   reccrdg 7075   1c1 9492    + caddc 9494   ZZcz 10863   ZZ>=cuz 11081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-n0 10795  df-z 10864  df-uz 11082
This theorem is referenced by:  seqp1  12089
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