Theorem islhp 39982

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 39981	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
6	5	eleq2d 2815	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }))
7		breq1 5118	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤𝐶 1 ↔ 𝑊𝐶 1 ))
8	7	elrab 3667	. 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 3411   class class class wbr 5115  ‘cfv 6519  Basecbs 17185  1.cp1 18389   ⋖ ccvr 39247  LHypclh 39970

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 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-lhyp 39974

This theorem is referenced by:  islhp2  39983  lhpbase  39984  lhp1cvr  39985