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Metamath Proof Explorer

Theorem lhpjat1

Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unity. (Contributed by NM, 18-May-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 lhpjat1 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ 𝑃 ) = 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 simpll ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝐾 ∈ HL )
7 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
8 7 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
9 8 ad2antlr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
10 simprl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑃 ∈ 𝐴 )
11 eqid ⊢ ( ⋖ ‘ 𝐾 ) = ( ⋖ ‘ 𝐾 )
12 3 11 5 lhp1cvr ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑊 ( ⋖ ‘ 𝐾 ) 1 )
13 12 adantr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → 𝑊 ( ⋖ ‘ 𝐾 ) 1 )
14 simprr ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ 𝑃 ≤ 𝑊 )
15 7 1 2 3 11 4 1cvrjat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ 𝐴 ) ∧ ( 𝑊 ( ⋖ ‘ 𝐾 ) 1 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ 𝑃 ) = 1 )
16 6 9 10 13 14 15 syl32anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑊 ∨ 𝑃 ) = 1 )