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Theorem om2uzsuci 12015
Description: The value of  G (see om2uz0i 12014) 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 4936 . . . 4  |-  ( z  =  A  ->  suc  z  =  suc  A )
21fveq2d 5861 . . 3  |-  ( z  =  A  ->  ( G `  suc  z )  =  ( G `  suc  A ) )
3 fveq2 5857 . . . 4  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
43oveq1d 6290 . . 3  |-  ( z  =  A  ->  (
( G `  z
)  +  1 )  =  ( ( G `
 A )  +  1 ) )
52, 4eqeq12d 2482 . 2  |-  ( z  =  A  ->  (
( G `  suc  z )  =  ( ( G `  z
)  +  1 )  <-> 
( G `  suc  A )  =  ( ( G `  A )  +  1 ) ) )
6 ovex 6300 . . 3  |-  ( ( G `  z )  +  1 )  e. 
_V
7 om2uz.2 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
8 oveq1 6282 . . . 4  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
9 oveq1 6282 . . . 4  |-  ( y  =  ( G `  z )  ->  (
y  +  1 )  =  ( ( G `
 z )  +  1 ) )
107, 8, 9frsucmpt2 7095 . . 3  |-  ( ( z  e.  om  /\  ( ( G `  z )  +  1 )  e.  _V )  ->  ( G `  suc  z )  =  ( ( G `  z
)  +  1 ) )
116, 10mpan2 671 . 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 1374    e. wcel 1762   _Vcvv 3106    |-> cmpt 4498   suc csuc 4873    |` cres 4994   ` cfv 5579  (class class class)co 6275   omcom 6671   reccrdg 7065   1c1 9482    + caddc 9484   ZZcz 10853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-om 6672  df-recs 7032  df-rdg 7066
This theorem is referenced by:  om2uzuzi  12016  om2uzlti  12017  om2uzrani  12019  om2uzrdg  12023  uzrdgsuci  12027  uzrdgxfr  12033  fzennn  12034  axdc4uzlem  12048  hashgadd  12400
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