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Theorem islhp 38345
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 38344 . . 3 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
65eleq2d 2824 . 2 (𝐾𝐴 → (𝑊𝐻𝑊 ∈ {𝑤𝐵𝑤𝐶 1 }))
7 breq1 5107 . . 3 (𝑤 = 𝑊 → (𝑤𝐶 1𝑊𝐶 1 ))
87elrab 3644 . 2 (𝑊 ∈ {𝑤𝐵𝑤𝐶 1 } ↔ (𝑊𝐵𝑊𝐶 1 ))
96, 8bitrdi 287 1 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  {crab 3406   class class class wbr 5104  cfv 6492  Basecbs 17018  1.cp1 18248  ccvr 37610  LHypclh 38333
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 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6444  df-fun 6494  df-fv 6500  df-lhyp 38337
This theorem is referenced by:  islhp2  38346  lhpbase  38347  lhp1cvr  38348
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