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Theorem lhp1cvr 34103
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 2606 . . 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 34100 . 2 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊 ∈ (Base‘𝐾) ∧ 𝑊𝐶 1 )))
65simplbda 651 1 ((𝐾𝐴𝑊𝐻) → 𝑊𝐶 1 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976   class class class wbr 4574  cfv 5787  Basecbs 15638  1.cp1 16804  ccvr 33367  LHypclh 34088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pr 4825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-iota 5751  df-fun 5789  df-fv 5795  df-lhyp 34092
This theorem is referenced by:  lhplt  34104  lhp2lt  34105  lhpexlt  34106  lhpexnle  34110  lhpjat1  34124  lhpmcvr  34127  cdlemb2  34145  lhpat  34147  dih1  35393
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