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Theorem islhp2 39474
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 39473 . 2 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
65baibd 538 1 ((𝐾𝐴𝑊𝐵) → (𝑊𝐻𝑊𝐶 1 ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098   class class class wbr 5150  cfv 6551  Basecbs 17185  1.cp1 18421  ccvr 38738  LHypclh 39461
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-lhyp 39465
This theorem is referenced by:  lhpoc  39491
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