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

Theorem om2uz0i 12058
Description: The mapping is a one-to-one mapping from onto upper integers that will be used to construct a recursive definition generator. Ordinal natural number 0 maps to complex number (normally 0 for the upper integers or 1 for the upper integers ), 1 maps to + 1, etc. This theorem shows the value of at ordinal natural number zero. (This series of theorems generalizes an earlier series for 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
om2uz.2
Assertion
Ref Expression
om2uz0i
Distinct variable group:   ,

Proof of Theorem om2uz0i
StepHypRef Expression
1 om2uz.2 . . 3
21fveq1i 5872 . 2
3 om2uz.1 . . 3
4 fr0g 7120 . . 3
53, 4ax-mp 5 . 2
62, 5eqtri 2486 1
Colors of variables: wff setvar class
Syntax hints:  =wceq 1395  e.wcel 1818   cvv 3109   c0 3784  e.cmpt 4510  |`cres 5006  `cfv 5593  (class class class)co 6296   com 6700  reccrdg 7094  1c1 9514   caddc 9516   cz 10889
This theorem is referenced by:  om2uzuzi  12060  om2uzrani  12063  om2uzrdg  12067  uzrdgxfr  12077  fzennn  12078  axdc4uzlem  12092  hashgadd  12445
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-om 6701  df-recs 7061  df-rdg 7095
  Copyright terms: Public domain W3C validator