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Theorem islhp 38354
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 38353 . . 3 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
65eleq2d 2823 . 2 (𝐾𝐴 → (𝑊𝐻𝑊 ∈ {𝑤𝐵𝑤𝐶 1 }))
7 breq1 5106 . . 3 (𝑤 = 𝑊 → (𝑤𝐶 1𝑊𝐶 1 ))
87elrab 3643 . 2 (𝑊 ∈ {𝑤𝐵𝑤𝐶 1 } ↔ (𝑊𝐵𝑊𝐶 1 ))
96, 8bitrdi 286 1 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {crab 3405   class class class wbr 5103  cfv 6491  Basecbs 17017  1.cp1 18247  ccvr 37619  LHypclh 38342
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5528  df-xp 5636  df-rel 5637  df-cnv 5638  df-co 5639  df-dm 5640  df-iota 6443  df-fun 6493  df-fv 6499  df-lhyp 38346
This theorem is referenced by:  islhp2  38355  lhpbase  38356  lhp1cvr  38357
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