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Metamath Proof Explorer

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 )