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Metamath Proof Explorer

Theorem lhpelim

Description: Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013)

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

Proof

Step Hyp Ref Expression
1 lhpelim.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 lhpelim.l ⊢ ≤ = ( le ‘ 𝐾 )
3 lhpelim.j ⊢ ∨ = ( join ‘ 𝐾 )
4 lhpelim.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 lhpelim.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 lhpelim.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
8 2 4 7 5 6 lhpmat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) )
9 8 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑃 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) )
10 9 oveq1d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∧ 𝑊 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( ( 0. ‘ 𝐾 ) ∨ ( 𝑋 ∧ 𝑊 ) ) )
11 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ HL )
12 simp2l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑃 ∈ 𝐴 )
13 11 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ Lat )
14 simp3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
15 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑊 ∈ 𝐻 )
16 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
17 15 16 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝑊 ∈ 𝐵 )
18 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
19 13 14 17 18 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
20 1 2 4 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
21 13 14 17 20 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 )
22 1 2 3 4 5 atmod4i2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑊 ) ≤ 𝑊 ) → ( ( 𝑃 ∧ 𝑊 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) )
23 11 12 19 17 21 22 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∧ 𝑊 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) )
24 hlol ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL )
25 11 24 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ OL )
26 1 3 7 olj02 ⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → ( ( 0. ‘ 𝐾 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑋 ∧ 𝑊 ) )
27 25 19 26 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 0. ‘ 𝐾 ) ∨ ( 𝑋 ∧ 𝑊 ) ) = ( 𝑋 ∧ 𝑊 ) )
28 10 23 27 3eqtr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑋 ∈ 𝐵 ) → ( ( 𝑃 ∨ ( 𝑋 ∧ 𝑊 ) ) ∧ 𝑊 ) = ( 𝑋 ∧ 𝑊 ) )