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Theorem uzrdgsuci 11255
Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 11251. (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 11253 . . . . 5  |-  S  Fn  ( ZZ>= `  C )
7 fnfun 5501 . . . . 5  |-  ( S  Fn  ( ZZ>= `  C
)  ->  Fun  S )
86, 7ax-mp 8 . . . 4  |-  Fun  S
9 peano2uz 10486 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
101, 2, 3, 4uzrdglem 11252 . . . . . 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 2495 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  S
)
13 funopfv 5725 . . . 4  |-  ( Fun 
S  ->  ( <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  S  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) ) )
148, 12, 13mpsyl 61 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 ( `' G `  ( B  +  1 ) ) ) ) )
151, 2om2uzf1oi 11248 . . . . . . . 8  |-  G : om
-1-1-onto-> ( ZZ>= `  C )
16 f1ocnvdm 5977 . . . . . . . 8  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
1715, 16mpan 652 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( `' G `  B )  e.  om )
18 peano2 4824 . . . . . . 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 11243 . . . . . . . 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 5974 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
2315, 22mpan 652 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  ( `' G `  B ) )  =  B )
2423oveq1d 6055 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  =  ( B  + 
1 ) )
2521, 24eqtrd 2436 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 ) )
26 f1ocnvfv 5975 . . . . . . 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 652 . . . . . 6  |-  ( suc  ( `' G `  B )  e.  om  ->  ( ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 )  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) ) )
2819, 25, 27sylc 58 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( `' G `  ( B  +  1 ) )  =  suc  ( `' G `  B ) )
2928fveq2d 5691 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( R `  ( `' G `  ( B  +  1
) ) )  =  ( R `  suc  ( `' G `  B ) ) )
3029fveq2d 5691 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
3114, 30eqtrd 2436 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
32 frsuc 6653 . . . . . . . 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 5688 . . . . . . . 8  |-  ( R `
 suc  ( `' G `  B )
)  =  ( ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om ) `  suc  ( `' G `  B ) )
344fveq1i 5688 . . . . . . . . 9  |-  ( R `
 ( `' G `  B ) )  =  ( ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  ( `' G `  B ) )
3534fveq2i 5690 . . . . . . . 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 2461 . . . . . . 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 11251 . . . . . . . . 9  |-  ( ( `' G `  B )  e.  om  ->  ( R `  ( `' G `  B )
)  =  <. ( G `  ( `' G `  B )
) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
3837fveq2d 5691 . . . . . . . 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 6043 . . . . . . . 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 2454 . . . . . . 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 2436 . . . . . 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 5701 . . . . . . 7  |-  ( G `
 ( `' G `  B ) )  e. 
_V
43 fvex 5701 . . . . . . 7  |-  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  _V
44 oveq1 6047 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z  +  1 )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
45 oveq1 6047 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z F w )  =  ( ( G `  ( `' G `  B ) ) F w ) )
4644, 45opeq12d 3952 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  <. ( z  +  1 ) ,  ( z F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >. )
47 oveq2 6048 . . . . . . . . 9  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  (
( G `  ( `' G `  B ) ) F w )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
4847opeq2d 3951 . . . . . . . 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 6047 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  +  1 )  =  ( z  +  1 ) )
50 oveq1 6047 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x F y )  =  ( z F y ) )
5149, 50opeq12d 3952 . . . . . . . . 9  |-  ( x  =  z  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( z  +  1 ) ,  ( z F y ) >. )
52 oveq2 6048 . . . . . . . . . 10  |-  ( y  =  w  ->  (
z F y )  =  ( z F w ) )
5352opeq2d 3951 . . . . . . . . 9  |-  ( y  =  w  ->  <. (
z  +  1 ) ,  ( z F y ) >.  =  <. ( z  +  1 ) ,  ( z F w ) >. )
5451, 53cbvmpt2v 6111 . . . . . . . 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 4387 . . . . . . . 8  |-  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  _V
5646, 48, 54, 55ovmpt2 6168 . . . . . . 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 654 . . . . . 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 2452 . . . . 5  |-  ( ( `' G `  B )  e.  om  ->  ( R `  suc  ( `' G `  B ) )  =  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
5958fveq2d 5691 . . . 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 6065 . . . . 5  |-  ( ( G `  ( `' G `  B ) )  +  1 )  e.  _V
61 ovex 6065 . . . . 5  |-  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e. 
_V
6260, 61op2nd 6315 . . . 4  |-  ( 2nd `  <. ( ( G `
 ( `' G `  B ) )  +  1 ) ,  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) >.
)  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )
6359, 62syl6eq 2452 . . 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 11252 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  ran  R )
6665, 5syl6eleqr 2495 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  S
)
67 funopfv 5725 . . . . 5  |-  ( Fun 
S  ->  ( <. B ,  ( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  S  ->  ( S `  B
)  =  ( 2nd `  ( R `  ( `' G `  B ) ) ) ) )
688, 66, 67mpsyl 61 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  B )  =  ( 2nd `  ( R `
 ( `' G `  B ) ) ) )
6968eqcomd 2409 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  =  ( S `  B ) )
7023, 69oveq12d 6058 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  =  ( B F ( S `  B ) ) )
7131, 64, 703eqtrd 2440 1  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( B F ( S `
 B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777    e. cmpt 4226   suc csuc 4543   omcom 4804   `'ccnv 4836   ran crn 4838    |` cres 4839   Fun wfun 5407    Fn wfn 5408   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   2ndc2nd 6307   reccrdg 6626   1c1 8947    + caddc 8949   ZZcz 10238   ZZ>=cuz 10444
This theorem is referenced by:  seqp1  11293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445
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