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Theorem islhp 37572
 Description: The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 11-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
islhp (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))

Proof of Theorem islhp
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 lhpset.b . . . 4 𝐵 = (Base‘𝐾)
2 lhpset.u . . . 4 1 = (1.‘𝐾)
3 lhpset.c . . . 4 𝐶 = ( ⋖ ‘𝐾)
4 lhpset.h . . . 4 𝐻 = (LHyp‘𝐾)
51, 2, 3, 4lhpset 37571 . . 3 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
65eleq2d 2837 . 2 (𝐾𝐴 → (𝑊𝐻𝑊 ∈ {𝑤𝐵𝑤𝐶 1 }))
7 breq1 5035 . . 3 (𝑤 = 𝑊 → (𝑤𝐶 1𝑊𝐶 1 ))
87elrab 3602 . 2 (𝑊 ∈ {𝑤𝐵𝑤𝐶 1 } ↔ (𝑊𝐵𝑊𝐶 1 ))
96, 8bitrdi 290 1 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074   class class class wbr 5032  ‘cfv 6335  Basecbs 16541  1.cp1 17714   ⋖ ccvr 36838  LHypclh 37560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-iota 6294  df-fun 6337  df-fv 6343  df-lhyp 37564 This theorem is referenced by:  islhp2  37573  lhpbase  37574  lhp1cvr  37575
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