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Theorem islhp 36072
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 36071 . . 3 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
65eleq2d 2893 . 2 (𝐾𝐴 → (𝑊𝐻𝑊 ∈ {𝑤𝐵𝑤𝐶 1 }))
7 breq1 4877 . . 3 (𝑤 = 𝑊 → (𝑤𝐶 1𝑊𝐶 1 ))
87elrab 3586 . 2 (𝑊 ∈ {𝑤𝐵𝑤𝐶 1 } ↔ (𝑊𝐵𝑊𝐶 1 ))
96, 8syl6bb 279 1 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1658  wcel 2166  {crab 3122   class class class wbr 4874  cfv 6124  Basecbs 16223  1.cp1 17392  ccvr 35338  LHypclh 36060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-iota 6087  df-fun 6126  df-fv 6132  df-lhyp 36064
This theorem is referenced by:  islhp2  36073  lhpbase  36074  lhp1cvr  36075
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