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 36291
Description: An atom covers zero. (atcv0 30033 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 2825 . . 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 36289 . 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 1530  wcel 2107   class class class wbr 5062  cfv 6351  Basecbs 16475  0.cp0 17639  ccvr 36265  Atomscatm 36266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-ats 36270
This theorem is referenced by:  0ltat  36294  leatb  36295  atnle0  36312  atlen0  36313  atcmp  36314  atcvreq0  36317  atcvr0eq  36429  lnnat  36430  athgt  36459  ps-2  36481  lhp0lt  37006
  Copyright terms: Public domain W3C validator