Metamath Proof Explorer


Theorem lhpjat2

Description: The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝐾 ∈ Lat )
8 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
9 8 4 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
10 9 ad2antrl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
11 8 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
12 11 ad2antlr ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 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 ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 𝑊 ) = ( 𝑊 𝑃 ) )
15 1 2 3 4 5 lhpjat1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑊 𝑃 ) = 1 )
16 14 15 eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 𝑊 ) = 1 )