MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  om2uz0i Unicode version

Theorem om2uz0i 11242
Description: The mapping  G is a one-to-one mapping from  om onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number  C (normally 0 for the upper integers  NN0 or 1 for the upper integers  NN), 1 maps to  C + 1, etc. This theorem shows the value of  G at ordinal natural number zero. (This series of theorems generalizes an earlier series for  NN0 contributed by Raph Levien, 10-Apr-2004.) (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
om2uz0i  |-  ( G `
 (/) )  =  C
Distinct variable group:    x, C
Allowed substitution hint:    G( x)

Proof of Theorem om2uz0i
StepHypRef Expression
1 om2uz.2 . . 3  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om )
21fveq1i 5688 . 2  |-  ( G `
 (/) )  =  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )
3 om2uz.1 . . 3  |-  C  e.  ZZ
4 fr0g 6652 . . 3  |-  ( C  e.  ZZ  ->  (
( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C )
53, 4ax-mp 8 . 2  |-  ( ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  C )  |`  om ) `  (/) )  =  C
62, 5eqtri 2424 1  |-  ( G `
 (/) )  =  C
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588    e. cmpt 4226   omcom 4804    |` cres 4839   ` cfv 5413  (class class class)co 6040   reccrdg 6626   1c1 8947    + caddc 8949   ZZcz 10238
This theorem is referenced by:  om2uzuzi  11244  om2uzrani  11247  om2uzrdg  11251  uzrdgxfr  11261  fzennn  11262  axdc4uzlem  11276  hashgadd  11606
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-recs 6592  df-rdg 6627
  Copyright terms: Public domain W3C validator