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Theorem om2uzsuci 12044
Description: The value of  G (see om2uz0i 12043) 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 4932 . . . 4  |-  ( z  =  A  ->  suc  z  =  suc  A )
21fveq2d 5852 . . 3  |-  ( z  =  A  ->  ( G `  suc  z )  =  ( G `  suc  A ) )
3 fveq2 5848 . . . 4  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
43oveq1d 6285 . . 3  |-  ( z  =  A  ->  (
( G `  z
)  +  1 )  =  ( ( G `
 A )  +  1 ) )
52, 4eqeq12d 2476 . 2  |-  ( z  =  A  ->  (
( G `  suc  z )  =  ( ( G `  z
)  +  1 )  <-> 
( G `  suc  A )  =  ( ( G `  A )  +  1 ) ) )
6 ovex 6298 . . 3  |-  ( ( G `  z )  +  1 )  e. 
_V
7 om2uz.2 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
8 oveq1 6277 . . . 4  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
9 oveq1 6277 . . . 4  |-  ( y  =  ( G `  z )  ->  (
y  +  1 )  =  ( ( G `
 z )  +  1 ) )
107, 8, 9frsucmpt2 7097 . . 3  |-  ( ( z  e.  om  /\  ( ( G `  z )  +  1 )  e.  _V )  ->  ( G `  suc  z )  =  ( ( G `  z
)  +  1 ) )
116, 10mpan2 669 . 2  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
125, 11vtoclga 3170 1  |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `
 A )  +  1 ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106    |-> cmpt 4497   suc csuc 4869    |` cres 4990   ` cfv 5570  (class class class)co 6270   omcom 6673   reccrdg 7067   1c1 9482    + caddc 9484   ZZcz 10860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-om 6674  df-recs 7034  df-rdg 7068
This theorem is referenced by:  om2uzuzi  12045  om2uzlti  12046  om2uzrani  12048  om2uzrdg  12052  uzrdgsuci  12056  uzrdgxfr  12062  fzennn  12063  axdc4uzlem  12077  hashgadd  12431
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