Theorem lhpjat2

Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unity. (Contributed by NM, 4-Jun-2012)

	Ref	Expression
Hypotheses	lhpjat.l	⊢ ≤ = ( le ‘ 𝐾 )
	lhpjat.j	⊢ ∨ = ( join ‘ 𝐾 )
	lhpjat.u	⊢ 1 = ( 1. ‘ 𝐾 )
	lhpjat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	lhpjat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion	lhpjat2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ 𝑊 ) = 1 )

Proof

Step	Hyp	Ref	Expression
1		lhpjat.l	⊢ ≤ = ( le ‘ 𝐾 )
2		lhpjat.j	⊢ ∨ = ( join ‘ 𝐾 )
3		lhpjat.u	⊢ 1 = ( 1. ‘ 𝐾 )
4		lhpjat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		lhpjat.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
7	6	ad2antrr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ Lat )
8		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9	8 4	atbase	⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
10	9	ad2antrl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
11	8 5	lhpbase	⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
12	11	ad2antlr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
13	8 2	latjcom	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑊 ) = ( 𝑊 ∨ 𝑃 ) )
14	7 10 12 13	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ 𝑊 ) = ( 𝑊 ∨ 𝑃 ) )
15	1 2 3 4 5	lhpjat1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ 𝑃 ) = 1 )
16	14 15	eqtrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ 𝑊 ) = 1 )