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Theorem om2uzsuci 11027
Description: The value of  G (see om2uz0i 11026) 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 4473 . . . 4  |-  ( z  =  A  ->  suc  z  =  suc  A )
21fveq2d 5545 . . 3  |-  ( z  =  A  ->  ( G `  suc  z )  =  ( G `  suc  A ) )
3 fveq2 5541 . . . 4  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
43oveq1d 5889 . . 3  |-  ( z  =  A  ->  (
( G `  z
)  +  1 )  =  ( ( G `
 A )  +  1 ) )
52, 4eqeq12d 2310 . 2  |-  ( z  =  A  ->  (
( G `  suc  z )  =  ( ( G `  z
)  +  1 )  <-> 
( G `  suc  A )  =  ( ( G `  A )  +  1 ) ) )
6 ovex 5899 . . 3  |-  ( ( G `  z )  +  1 )  e. 
_V
7 om2uz.2 . . . 4  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
8 oveq1 5881 . . . 4  |-  ( y  =  x  ->  (
y  +  1 )  =  ( x  + 
1 ) )
9 oveq1 5881 . . . 4  |-  ( y  =  ( G `  z )  ->  (
y  +  1 )  =  ( ( G `
 z )  +  1 ) )
107, 8, 9frsucmpt2 6468 . . 3  |-  ( ( z  e.  om  /\  ( ( G `  z )  +  1 )  e.  _V )  ->  ( G `  suc  z )  =  ( ( G `  z
)  +  1 ) )
116, 10mpan2 652 . 2  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
125, 11vtoclga 2862 1  |-  ( A  e.  om  ->  ( G `  suc  A )  =  ( ( G `
 A )  +  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   suc csuc 4410   omcom 4672    |` cres 4707   ` cfv 5271  (class class class)co 5874   reccrdg 6438   1c1 8754    + caddc 8756   ZZcz 10040
This theorem is referenced by:  om2uzuzi  11028  om2uzlti  11029  om2uzrani  11031  om2uzrdg  11035  uzrdgsuci  11039  uzrdgxfr  11045  fzennn  11046  axdc4uzlem  11060  hashgadd  11375
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-recs 6404  df-rdg 6439
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