Metamath Proof Explorer


Theorem lhple

Description: Property of a lattice element under a co-atom. (Contributed by NM, 28-Feb-2014)

Ref Expression
Hypotheses lhple.b 𝐵 = ( Base ‘ 𝐾 )
lhple.l = ( le ‘ 𝐾 )
lhple.j = ( join ‘ 𝐾 )
lhple.m = ( meet ‘ 𝐾 )
lhple.a 𝐴 = ( Atoms ‘ 𝐾 )
lhple.h 𝐻 = ( LHyp ‘ 𝐾 )
Assertion lhple ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ( 𝑃 𝑋 ) 𝑊 ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 lhple.b 𝐵 = ( Base ‘ 𝐾 )
2 lhple.l = ( le ‘ 𝐾 )
3 lhple.j = ( join ‘ 𝐾 )
4 lhple.m = ( meet ‘ 𝐾 )
5 lhple.a 𝐴 = ( Atoms ‘ 𝐾 )
6 lhple.h 𝐻 = ( LHyp ‘ 𝐾 )
7 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → 𝐾 ∈ HL )
8 7 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → 𝐾 ∈ Lat )
9 simp2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → 𝑃𝐴 )
10 1 5 atbase ( 𝑃𝐴𝑃𝐵 )
11 9 10 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → 𝑃𝐵 )
12 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → 𝑋𝐵 )
13 1 3 latjcom ( ( 𝐾 ∈ Lat ∧ 𝑃𝐵𝑋𝐵 ) → ( 𝑃 𝑋 ) = ( 𝑋 𝑃 ) )
14 8 11 12 13 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝑃 𝑋 ) = ( 𝑋 𝑃 ) )
15 14 oveq1d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ( 𝑃 𝑋 ) 𝑊 ) = ( ( 𝑋 𝑃 ) 𝑊 ) )
16 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → 𝑋 𝑊 )
18 1 2 3 4 6 lhpmod6i1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑋𝐵𝑃𝐵 ) ∧ 𝑋 𝑊 ) → ( 𝑋 ( 𝑃 𝑊 ) ) = ( ( 𝑋 𝑃 ) 𝑊 ) )
19 16 12 11 17 18 syl121anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝑋 ( 𝑃 𝑊 ) ) = ( ( 𝑋 𝑃 ) 𝑊 ) )
20 eqid ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
21 2 4 20 5 6 lhpmat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 𝑊 ) = ( 0. ‘ 𝐾 ) )
22 21 3adant3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝑃 𝑊 ) = ( 0. ‘ 𝐾 ) )
23 22 oveq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝑋 ( 𝑃 𝑊 ) ) = ( 𝑋 ( 0. ‘ 𝐾 ) ) )
24 hlol ( 𝐾 ∈ HL → 𝐾 ∈ OL )
25 7 24 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → 𝐾 ∈ OL )
26 1 3 20 olj01 ( ( 𝐾 ∈ OL ∧ 𝑋𝐵 ) → ( 𝑋 ( 0. ‘ 𝐾 ) ) = 𝑋 )
27 25 12 26 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝑋 ( 0. ‘ 𝐾 ) ) = 𝑋 )
28 23 27 eqtrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( 𝑋 ( 𝑃 𝑊 ) ) = 𝑋 )
29 15 19 28 3eqtr2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑋𝐵𝑋 𝑊 ) ) → ( ( 𝑃 𝑋 ) 𝑊 ) = 𝑋 )