Theorem islhp 40293

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 40292	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
6	5	eleq2d 2823	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }))
7		breq1 5102	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤𝐶 1 ↔ 𝑊𝐶 1 ))
8	7	elrab 3647	. 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 1542   ∈ wcel 2114  {crab 3400   class class class wbr 5099  ‘cfv 6493  Basecbs 17140  1.cp1 18349   ⋖ ccvr 39559  LHypclh 40281

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-lhyp 40285

This theorem is referenced by:  islhp2  40294  lhpbase  40295  lhp1cvr  40296