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Theorem om2uzsuci 11216
Description: The value of  G (see om2uz0i 11215) at a successor. (Contributed by NM, 3-Oct-2004.) (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 )
Assertion
Ref Expression
om2uzsuci  |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `
 A )  +  1 ) )
Distinct variable group:    x, C
Allowed substitution hints:    A( x)    G( x)

Proof of Theorem om2uzsuci
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suceq 4588 . . . 4  |-  ( z  =  A  ->  suc  z  =  suc  A )
21fveq2d 5673 . . 3  |-  ( z  =  A  ->  ( G `  suc  z )  =  ( G `  suc  A ) )
3 fveq2 5669 . . . 4  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
43oveq1d 6036 . . 3  |-  ( z  =  A  ->  (
( G `  z
)  +  1 )  =  ( ( G `
 A )  +  1 ) )
52, 4eqeq12d 2402 . 2  |-  ( z  =  A  ->  (
( G `  suc  z )  =  ( ( G `  z
)  +  1 )  <-> 
( G `  suc  A )  =  ( ( G `  A )  +  1 ) ) )
6 ovex 6046 . . 3  |-  ( ( G `  z )  +  1 )  e. 
_V
7 om2uz.2 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
8 oveq1 6028 . . . 4  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
9 oveq1 6028 . . . 4  |-  ( y  =  ( G `  z )  ->  (
y  +  1 )  =  ( ( G `
 z )  +  1 ) )
107, 8, 9frsucmpt2 6634 . . 3  |-  ( ( z  e.  om  /\  ( ( G `  z )  +  1 )  e.  _V )  ->  ( G `  suc  z )  =  ( ( G `  z
)  +  1 ) )
116, 10mpan2 653 . 2  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
125, 11vtoclga 2961 1  |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `
 A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2900    e. cmpt 4208   suc csuc 4525   omcom 4786    |` cres 4821   ` cfv 5395  (class class class)co 6021   reccrdg 6604   1c1 8925    + caddc 8927   ZZcz 10215
This theorem is referenced by:  om2uzuzi  11217  om2uzlti  11218  om2uzrani  11220  om2uzrdg  11224  uzrdgsuci  11228  uzrdgxfr  11234  fzennn  11235  axdc4uzlem  11249  hashgadd  11579
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-recs 6570  df-rdg 6605
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