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Theorem uzrdgsuci 11767
Description: Successor value of a recursive definition generator on upper integers. See comment in om2uzrdg 11763. (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 11765 . . . . 5  |-  S  Fn  ( ZZ>= `  C )
7 fnfun 5496 . . . . 5  |-  ( S  Fn  ( ZZ>= `  C
)  ->  Fun  S )
86, 7ax-mp 5 . . . 4  |-  Fun  S
9 peano2uz 10896 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( B  +  1 )  e.  ( ZZ>= `  C )
)
101, 2, 3, 4uzrdglem 11764 . . . . . 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 2524 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. ( B  +  1 ) ,  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) ) >.  e.  S
)
13 funopfv 5719 . . . 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 11760 . . . . . . . 8  |-  G : om
-1-1-onto-> ( ZZ>= `  C )
16 f1ocnvdm 5976 . . . . . . . 8  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( `' G `  B )  e.  om )
1715, 16mpan 663 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( `' G `  B )  e.  om )
18 peano2 6485 . . . . . . 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 11755 . . . . . . . 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 5971 . . . . . . . . 9  |-  ( ( G : om -1-1-onto-> ( ZZ>= `  C )  /\  B  e.  ( ZZ>=
`  C ) )  ->  ( G `  ( `' G `  B ) )  =  B )
2315, 22mpan 663 . . . . . . . 8  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  ( `' G `  B ) )  =  B )
2423oveq1d 6095 . . . . . . 7  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
)  +  1 )  =  ( B  + 
1 ) )
2521, 24eqtrd 2465 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( G `  suc  ( `' G `  B ) )  =  ( B  +  1 ) )
26 f1ocnvfv 5972 . . . . . . 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 663 . . . . . 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 5683 . . . 4  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( R `  ( `' G `  ( B  +  1
) ) )  =  ( R `  suc  ( `' G `  B ) ) )
3029fveq2d 5683 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  ( B  +  1 ) ) ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
3114, 30eqtrd 2465 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( 2nd `  ( R `
 suc  ( `' G `  B )
) ) )
32 frsuc 6878 . . . . . . . 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 5680 . . . . . . . 8  |-  ( R `
 suc  ( `' G `  B )
)  =  ( ( rec ( ( x  e.  _V ,  y  e.  _V  |->  <. (
x  +  1 ) ,  ( x F y ) >. ) ,  <. C ,  A >. )  |`  om ) `  suc  ( `' G `  B ) )
344fveq1i 5680 . . . . . . . . 9  |-  ( R `
 ( `' G `  B ) )  =  ( ( rec (
( x  e.  _V ,  y  e.  _V  |->  <. ( x  +  1 ) ,  ( x F y ) >.
) ,  <. C ,  A >. )  |`  om ) `  ( `' G `  B ) )
3534fveq2i 5682 . . . . . . . 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 2490 . . . . . . 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 11763 . . . . . . . . 9  |-  ( ( `' G `  B )  e.  om  ->  ( R `  ( `' G `  B )
)  =  <. ( G `  ( `' G `  B )
) ,  ( 2nd `  ( R `  ( `' G `  B ) ) ) >. )
3837fveq2d 5683 . . . . . . . 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 6083 . . . . . . . 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 2483 . . . . . . 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 2465 . . . . . 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 5689 . . . . . . 7  |-  ( G `
 ( `' G `  B ) )  e. 
_V
43 fvex 5689 . . . . . . 7  |-  ( 2nd `  ( R `  ( `' G `  B ) ) )  e.  _V
44 oveq1 6087 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z  +  1 )  =  ( ( G `  ( `' G `  B ) )  +  1 ) )
45 oveq1 6087 . . . . . . . . 9  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  ( z F w )  =  ( ( G `  ( `' G `  B ) ) F w ) )
4644, 45opeq12d 4055 . . . . . . . 8  |-  ( z  =  ( G `  ( `' G `  B ) )  ->  <. ( z  +  1 ) ,  ( z F w ) >.  =  <. ( ( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F w ) >. )
47 oveq2 6088 . . . . . . . . 9  |-  ( w  =  ( 2nd `  ( R `  ( `' G `  B )
) )  ->  (
( G `  ( `' G `  B ) ) F w )  =  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) )
4847opeq2d 4054 . . . . . . . 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 6087 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x  +  1 )  =  ( z  +  1 ) )
50 oveq1 6087 . . . . . . . . . 10  |-  ( x  =  z  ->  (
x F y )  =  ( z F y ) )
5149, 50opeq12d 4055 . . . . . . . . 9  |-  ( x  =  z  ->  <. (
x  +  1 ) ,  ( x F y ) >.  =  <. ( z  +  1 ) ,  ( z F y ) >. )
52 oveq2 6088 . . . . . . . . . 10  |-  ( y  =  w  ->  (
z F y )  =  ( z F w ) )
5352opeq2d 4054 . . . . . . . . 9  |-  ( y  =  w  ->  <. (
z  +  1 ) ,  ( z F y ) >.  =  <. ( z  +  1 ) ,  ( z F w ) >. )
5451, 53cbvmpt2v 6155 . . . . . . . 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 4544 . . . . . . . 8  |-  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.  e.  _V
5646, 48, 54, 55ovmpt2 6215 . . . . . . 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 665 . . . . . 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 2481 . . . . 5  |-  ( ( `' G `  B )  e.  om  ->  ( R `  suc  ( `' G `  B ) )  =  <. (
( G `  ( `' G `  B ) )  +  1 ) ,  ( ( G `
 ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B )
) ) ) >.
)
5958fveq2d 5683 . . . 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 6105 . . . . 5  |-  ( ( G `  ( `' G `  B ) )  +  1 )  e.  _V
61 ovex 6105 . . . . 5  |-  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  e. 
_V
6260, 61op2nd 6575 . . . 4  |-  ( 2nd `  <. ( ( G `
 ( `' G `  B ) )  +  1 ) ,  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) ) >.
)  =  ( ( G `  ( `' G `  B ) ) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )
6359, 62syl6eq 2481 . . 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 11764 . . . . . 6  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  ran  R )
6665, 5syl6eleqr 2524 . . . . 5  |-  ( B  e.  ( ZZ>= `  C
)  ->  <. B , 
( 2nd `  ( R `  ( `' G `  B )
) ) >.  e.  S
)
67 funopfv 5719 . . . . 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 2438 . . 3  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( 2nd `  ( R `  ( `' G `  B ) ) )  =  ( S `  B ) )
7023, 69oveq12d 6098 . 2  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( ( G `  ( `' G `  B )
) F ( 2nd `  ( R `  ( `' G `  B ) ) ) )  =  ( B F ( S `  B ) ) )
7131, 64, 703eqtrd 2469 1  |-  ( B  e.  ( ZZ>= `  C
)  ->  ( S `  ( B  +  1 ) )  =  ( B F ( S `
 B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1362    e. wcel 1755   _Vcvv 2962   <.cop 3871    e. cmpt 4338   suc csuc 4708   `'ccnv 4826   ran crn 4828    |` cres 4829   Fun wfun 5400    Fn wfn 5401   -1-1-onto->wf1o 5405   ` cfv 5406  (class class class)co 6080    e. cmpt2 6082   omcom 6465   2ndc2nd 6565   reccrdg 6851   1c1 9271    + caddc 9273   ZZcz 10634   ZZ>=cuz 10849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-2nd 6567  df-recs 6818  df-rdg 6852  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-n0 10568  df-z 10635  df-uz 10850
This theorem is referenced by:  seqp1  11805
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