Theorem lhpat4N

Description: Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013) (New usage is discouraged.)

	Ref	Expression
Hypotheses	lhpat.l	⊢ ≤ = ( le ‘ 𝐾 )
	lhpat.j	⊢ ∨ = ( join ‘ 𝐾 )
	lhpat.m	⊢ ∧ = ( meet ‘ 𝐾 )
	lhpat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpat4N	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ 𝑈 ) ∧ 𝑊 ) = 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		lhpat.l	⊢ ≤ = ( le ‘ 𝐾 )
2		lhpat.j	⊢ ∨ = ( join ‘ 𝐾 )
3		lhpat.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		lhpat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		lhpat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7		simp2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
8		simp3l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → 𝑈 ∈ 𝐴 )
9		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10	9 4	atbase	⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
11	8 10	syl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
12		simp3r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → 𝑈 ≤ 𝑊 )
13	9 1 2 3 4 5	lhple	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ 𝑈 ) ∧ 𝑊 ) = 𝑈 )
14	6 7 11 12 13	syl112anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊 ) ) → ( ( 𝑃 ∨ 𝑈 ) ∧ 𝑊 ) = 𝑈 )