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Theorem islhp2 38506
Description: The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 18-May-2012.)
Hypotheses
Ref Expression
lhpset.b 𝐵 = (Base‘𝐾)
lhpset.u 1 = (1.‘𝐾)
lhpset.c 𝐶 = ( ⋖ ‘𝐾)
lhpset.h 𝐻 = (LHyp‘𝐾)
Assertion
Ref Expression
islhp2 ((𝐾𝐴𝑊𝐵) → (𝑊𝐻𝑊𝐶 1 ))

Proof of Theorem islhp2
StepHypRef Expression
1 lhpset.b . . 3 𝐵 = (Base‘𝐾)
2 lhpset.u . . 3 1 = (1.‘𝐾)
3 lhpset.c . . 3 𝐶 = ( ⋖ ‘𝐾)
4 lhpset.h . . 3 𝐻 = (LHyp‘𝐾)
51, 2, 3, 4islhp 38505 . 2 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
65baibd 541 1 ((𝐾𝐴𝑊𝐵) → (𝑊𝐻𝑊𝐶 1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107   class class class wbr 5106  cfv 6497  Basecbs 17088  1.cp1 18318  ccvr 37770  LHypclh 38493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-lhyp 38497
This theorem is referenced by:  lhpoc  38523
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