Theorem islhp 41129

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 41128	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐻 = {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 })
6	5	eleq2d 2846	. 2 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 }))
7		breq1 5105	. . 3 ⊢ (𝑤 = 𝑊 → (𝑤𝐶 1 ↔ 𝑊𝐶 1 ))
8	7	elrab 3644	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ 𝐵 ∣ 𝑤𝐶 1 } ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 ))
9	6, 8	bitrdi 290	1 ⊢ (𝐾 ∈ 𝐴 → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ 𝐵 ∧ 𝑊𝐶 1 )))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412   class class class wbr 5102  ‘cfv 6538  Basecbs 17388  1.cp1 18597   ⋖ ccvr 40395  LHypclh 41117

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-lhyp 41121

This theorem is used by:  islhp2  41130  lhpbase  41131  lhp1cvr  41132