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Metamath Proof Explorer

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 ) )