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Theorem lhp1cvr 37015
Description: The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
lhp1cvr.u 1 = (1.‘𝐾)
lhp1cvr.c 𝐶 = ( ⋖ ‘𝐾)
lhp1cvr.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
lhp1cvr ((𝐾𝐴𝑊𝐻) → 𝑊𝐶 1 )

Proof of Theorem lhp1cvr
StepHypRef Expression
1 eqid 2818 . . 3 (Base‘𝐾) = (Base‘𝐾)
2 lhp1cvr.u . . 3 1 = (1.‘𝐾)
3 lhp1cvr.c . . 3 𝐶 = ( ⋖ ‘𝐾)
4 lhp1cvr.h . . 3 𝐻 = (LHyp‘𝐾)
51, 2, 3, 4islhp 37012 . 2 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊𝐶 1 )))
65simplbda 500 1 ((𝐾𝐴𝑊𝐻) → 𝑊𝐶 1 )
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  1.cp1 17636  ccvr 36278  LHypclh 37000
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-lhyp 37004
This theorem is referenced by:  lhplt  37016  lhp2lt  37017  lhpexlt  37018  lhpexnle  37022  lhpjat1  37036  lhpmcvr  37039  cdlemb2  37057  lhpat  37059  dih1  38302
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