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Metamath Proof Explorer

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 )