Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  atcvr0 Structured version   Visualization version   GIF version

Theorem atcvr0 36304
Description: An atom covers zero. (atcv0 30046 analog.) (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atomcvr0.z 0 = (0.‘𝐾)
atomcvr0.c 𝐶 = ( ⋖ ‘𝐾)
atomcvr0.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
atcvr0 ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)

Proof of Theorem atcvr0
StepHypRef Expression
1 eqid 2818 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 atomcvr0.z . . 3 0 = (0.‘𝐾)
3 atomcvr0.c . . 3 𝐶 = ( ⋖ ‘𝐾)
4 atomcvr0.a . . 3 𝐴 = (Atoms‘𝐾)
51, 2, 3, 4isat 36302 . 2 (𝐾𝐷 → (𝑃𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 0 𝐶𝑃)))
65simplbda 500 1 ((𝐾𝐷𝑃𝐴) → 0 𝐶𝑃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  Basecbs 16471  0.cp0 17635  ccvr 36278  Atomscatm 36279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ats 36283
This theorem is referenced by:  0ltat  36307  leatb  36308  atnle0  36325  atlen0  36326  atcmp  36327  atcvreq0  36330  atcvr0eq  36442  lnnat  36443  athgt  36472  ps-2  36494  lhp0lt  37019
  Copyright terms: Public domain W3C validator