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Theorem lhp1cvr 33982
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  =  (  <o  `  K )
lhp1cvr.h  |-  H  =  ( LHyp `  K
)
Assertion
Ref Expression
lhp1cvr  |-  ( ( K  e.  A  /\  W  e.  H )  ->  W C  .1.  )

Proof of Theorem lhp1cvr
StepHypRef Expression
1 eqid 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
2 lhp1cvr.u . . 3  |-  .1.  =  ( 1. `  K )
3 lhp1cvr.c . . 3  |-  C  =  (  <o  `  K )
4 lhp1cvr.h . . 3  |-  H  =  ( LHyp `  K
)
51, 2, 3, 4islhp 33979 . 2  |-  ( K  e.  A  ->  ( W  e.  H  <->  ( W  e.  ( Base `  K
)  /\  W C  .1.  ) ) )
65simplbda 624 1  |-  ( ( K  e.  A  /\  W  e.  H )  ->  W C  .1.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527   Basecbs 14293   1.cp1 15328    <o ccvr 33246   LHypclh 33967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-lhyp 33971
This theorem is referenced by:  lhplt  33983  lhp2lt  33984  lhpexlt  33985  lhpexnle  33989  lhpjat1  34003  lhpmcvr  34006  cdlemb2  34024  lhpat  34026  dih1  35270
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