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Metamath Proof Explorer

Theorem lhpmcvr6N

Description: Specialization of lhpmcvr2 . (Contributed by NM, 6-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses lhpmcvr2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
lhpmcvr2.l ⊢ ≤ = ( le ‘ 𝐾 )
lhpmcvr2.j ⊢ ∨ = ( join ‘ 𝐾 )
lhpmcvr2.m ⊢ ∧ = ( meet ‘ 𝐾 )
lhpmcvr2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
lhpmcvr2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion lhpmcvr6N ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋 ) )

Proof

Step Hyp Ref Expression
1 lhpmcvr2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 lhpmcvr2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 lhpmcvr2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 lhpmcvr2.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 lhpmcvr2.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 lhpmcvr2.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 1 2 3 4 5 6 lhpmcvr5N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
8 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ¬ 𝑝 ≤ 𝑊 )
9 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ¬ 𝑝 ≤ 𝑌 )
10 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐾 ∈ HL )
11 10 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐾 ∈ Lat )
12 1 5 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
13 12 3ad2ant2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑝 ∈ 𝐵 )
14 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑋 ∈ 𝐵 )
15 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑊 ∈ 𝐻 )
16 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
17 15 16 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑊 ∈ 𝐵 )
18 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
19 11 14 17 18 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
20 1 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → 𝑝 ≤ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) )
21 11 13 19 20 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑝 ≤ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) )
22 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
23 21 22 breqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑝 ≤ 𝑋 )
24 8 9 23 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋 ) )
25 24 3expia ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋 ) ) )
26 25 reximdva ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋 ) ) )
27 7 26 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ 𝑝 ≤ 𝑋 ) )