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Theorem uzrdgsuci 12212
Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 12208. (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 12210 . . . . 5  |-  S  Fn  ( ZZ>= `  C )
7 fnfun 5683 . . . . 5  |-  ( S  Fn  ( ZZ>= `  C
)  ->  Fun  S )
86, 7ax-mp 5 . . . 4  |-  Fun  S
9 peano2uz 11235 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
101, 2, 3, 4uzrdglem 12209 . . . . . 6  |-  ( ( B  +  1 )  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  ran  R )
119, 10syl 17 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  ran  R )
1211, 5syl6eleqr 2560 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  S
)
13 funopfv 5918 . . . 4  |-  ( Fun 
S  ->  ( <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  S  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
148, 12, 13mpsyl 64 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 ( `' G `  ( B  +  1 ) ) ) ) )
151, 2om2uzf1oi 12205 . . . . . . . 8  |-  G : om
-1-1-onto-> ( ZZ>= `  C )
16 f1ocnvdm 6201 . . . . . . . 8  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
1715, 16mpan 684 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( `' G `  B )  e.  om )
18 peano2 6732 . . . . . . 7  |-  ( ( `' G `  B )  e.  om  ->  suc  ( `' G `  B )  e.  om )
1917, 18syl 17 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  suc  ( `' G `  B )  e.  om )
201, 2om2uzsuci 12200 . . . . . . . 8  |-  ( ( `' G `  B )  e.  om  ->  ( G `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
2117, 20syl 17 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
22 f1ocnvfv2 6194 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
2315, 22mpan 684 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  ( `' G `  B ) )  =  B )
2423oveq1d 6323 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  =  ( B  + 
1 ) )
2521, 24eqtrd 2505 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 ) )
26 f1ocnvfv 6195 . . . . . . 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 684 . . . . . 6  |-  ( suc  ( `' G `  B )  e.  om  ->  ( ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 )  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) ) )
2819, 25, 27sylc 61 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) )
2928fveq2d 5883 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( R `  ( `' G `  ( B  +  1
) ) )  =  ( R `  suc  ( `' G `  B ) ) )
3029fveq2d 5883 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
3114, 30eqtrd 2505 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
32 frsuc 7172 . . . . . . . 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 5880 . . . . . . . 8  |-  ( R `
 suc  ( `' G `  B )
)  =  ( ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om ) `  suc  ( `' G `  B ) )
344fveq1i 5880 . . . . . . . . 9  |-  ( R `
 ( `' G `  B ) )  =  ( ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  ( `' G `  B ) )
3534fveq2i 5882 . . . . . . . 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 2530 . . . . . . 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 12208 . . . . . . . . 9  |-  ( ( `' G `  B )  e.  om  ->  ( R `  ( `' G `  B )
)  =  <. ( G `  ( `' G `  B )
) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
3837fveq2d 5883 . . . . . . . 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 6311 . . . . . . . 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 2523 . . . . . . 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 2505 . . . . . 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 5889 . . . . . . 7  |-  ( G `
 ( `' G `  B ) )  e. 
_V
43 fvex 5889 . . . . . . 7  |-  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  _V
44 oveq1 6315 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z  +  1 )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
45 oveq1 6315 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z F w )  =  ( ( G `  ( `' G `  B ) ) F w ) )
4644, 45opeq12d 4166 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  <. ( z  +  1 ) ,  ( z F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >. )
47 oveq2 6316 . . . . . . . . 9  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  (
( G `  ( `' G `  B ) ) F w )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
4847opeq2d 4165 . . . . . . . 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 6315 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  +  1 )  =  ( z  +  1 ) )
50 oveq1 6315 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x F y )  =  ( z F y ) )
5149, 50opeq12d 4166 . . . . . . . . 9  |-  ( x  =  z  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( z  +  1 ) ,  ( z F y ) >. )
52 oveq2 6316 . . . . . . . . . 10  |-  ( y  =  w  ->  (
z F y )  =  ( z F w ) )
5352opeq2d 4165 . . . . . . . . 9  |-  ( y  =  w  ->  <. (
z  +  1 ) ,  ( z F y ) >.  =  <. ( z  +  1 ) ,  ( z F w ) >. )
5451, 53cbvmpt2v 6390 . . . . . . . 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 4664 . . . . . . . 8  |-  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  _V
5646, 48, 54, 55ovmpt2 6451 . . . . . . 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 686 . . . . . 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 2521 . . . . 5  |-  ( ( `' G `  B )  e.  om  ->  ( R `  suc  ( `' G `  B ) )  =  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
5958fveq2d 5883 . . . 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 6336 . . . . 5  |-  ( ( G `  ( `' G `  B ) )  +  1 )  e.  _V
61 ovex 6336 . . . . 5  |-  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e. 
_V
6260, 61op2nd 6821 . . . 4  |-  ( 2nd `  <. ( ( G `
 ( `' G `  B ) )  +  1 ) ,  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) >.
)  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )
6359, 62syl6eq 2521 . . 3  |-  ( ( `' G `  B )  e.  om  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
6417, 63syl 17 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  suc  ( `' G `  B ) ) )  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
651, 2, 3, 4uzrdglem 12209 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  ran  R )
6665, 5syl6eleqr 2560 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  S
)
67 funopfv 5918 . . . . 5  |-  ( Fun 
S  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  S  ->  ( S `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
688, 66, 67mpsyl 64 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  B )  =  ( 2nd `  ( R `
 ( `' G `  B ) ) ) )
6968eqcomd 2477 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  =  ( S `  B ) )
7023, 69oveq12d 6326 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  =  ( B F ( S `  B ) ) )
7131, 64, 703eqtrd 2509 1  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( B F ( S `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904   _Vcvv 3031   <.cop 3965    |-> cmpt 4454   `'ccnv 4838   ran crn 4840    |` cres 4841   suc csuc 5432   Fun wfun 5583    Fn wfn 5584   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   omcom 6711   2ndc2nd 6811   reccrdg 7145   1c1 9558    + caddc 9560   ZZcz 10961   ZZ>=cuz 11182
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183
This theorem is referenced by:  seqp1  12266
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