Theorem islhp2 37132

Description: The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 18-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
islhp2	⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ 𝑊𝐶 1 ))

Proof of Theorem islhp2

Step	Hyp	Ref	Expression
1		lhpset.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		lhpset.u	. . 3 ⊢ 1 = (1.‘𝐾)
3		lhpset.c	. . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾)
4		lhpset.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
5	1, 2, 3, 4	islhp 37131	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 )))
6	5	baibd 542	1 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑊 ∈ 𝐵) → (𝑊 ∈ 𝐻 ↔ 𝑊𝐶 1 ))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   class class class wbr 5065  ‘cfv 6354  Basecbs 16482  1.cp1 17647   ⋖ ccvr 36397  LHypclh 37119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-lhyp 37123

This theorem is referenced by:  lhpoc  37149