Theorem islhp 39995

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	⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 )))

Proof of Theorem islhp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	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)
5	1, 2, 3, 4	lhpset 39994	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
6	5	eleq2d 2814	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }))
7		breq1 5095	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤𝐶 1 ↔ 𝑊𝐶 1 ))
8	7	elrab 3648	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 ))
9	6, 8	bitrdi 287	1 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3394   class class class wbr 5092  ‘cfv 6482  Basecbs 17120  1.cp1 18328   ⋖ ccvr 39261  LHypclh 39983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6438  df-fun 6484  df-fv 6490  df-lhyp 39987

This theorem is referenced by:  islhp2  39996  lhpbase  39997  lhp1cvr  39998