Theorem lhpmcvr2

Description: Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013)

	Ref	Expression
Hypotheses	lhpmcvr2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	lhpmcvr2.l	⊢ ≤ = ( le ‘ 𝐾 )
	lhpmcvr2.j	⊢ ∨ = ( join ‘ 𝐾 )
	lhpmcvr2.m	⊢ ∧ = ( meet ‘ 𝐾 )
	lhpmcvr2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpmcvr2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpmcvr2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		lhpmcvr2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		lhpmcvr2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		lhpmcvr2.j	⊢ ∨ = ( join ‘ 𝐾 )
4		lhpmcvr2.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		lhpmcvr2.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		lhpmcvr2.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		eqid	⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
8	1 2 4 7 6	lhpmcvr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( 𝑋 ∧ 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 )
9		simpll	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
10		simprl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝑋 ∈ 𝐵 )
11	1 6	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
12	11	ad2antlr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝑊 ∈ 𝐵 )
13	1 2 3 4 7 5	cvrval5	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( ( 𝑋 ∧ 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) )
14	9 10 12 13	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ( 𝑋 ∧ 𝑊 ) ( ⋖ ‘ 𝐾 ) 𝑋 ↔ ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) )
15	8 14	mpbid	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )