Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islhp Structured version   Visualization version   GIF version

Theorem islhp 34083
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 34082 . . 3 (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })
65eleq2d 2672 . 2 (𝐾𝐴 → (𝑊𝐻𝑊 ∈ {𝑤𝐵𝑤𝐶 1 }))
7 breq1 4580 . . 3 (𝑤 = 𝑊 → (𝑤𝐶 1𝑊𝐶 1 ))
87elrab 3330 . 2 (𝑊 ∈ {𝑤𝐵𝑤𝐶 1 } ↔ (𝑊𝐵𝑊𝐶 1 ))
96, 8syl6bb 274 1 (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  {crab 2899   class class class wbr 4577  cfv 5789  Basecbs 15643  1.cp1 16809  ccvr 33350  LHypclh 34071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-iota 5753  df-fun 5791  df-fv 5797  df-lhyp 34075
This theorem is referenced by:  islhp2  34084  lhpbase  34085  lhp1cvr  34086
  Copyright terms: Public domain W3C validator