Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lhp1cvr Unicode version

Theorem lhp1cvr 30810
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 2296 . . 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 30807 . 2  |-  ( K  e.  A  ->  ( W  e.  H  <->  ( W  e.  ( Base `  K
)  /\  W C  .1.  ) ) )
65simplbda 607 1  |-  ( ( K  e.  A  /\  W  e.  H )  ->  W C  .1.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271   Basecbs 13164   1.cp1 14160    <o ccvr 30074   LHypclh 30795
This theorem is referenced by:  lhplt  30811  lhp2lt  30812  lhpexlt  30813  lhpexnle  30817  lhpjat1  30831  lhpmcvr  30834  cdlemb2  30852  lhpat  30854  dih1  32098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-lhyp 30799
  Copyright terms: Public domain W3C validator