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Metamath Proof Explorer

Theorem lhpmcvr5N

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 lhpmcvr5N ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 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 lhpmcvr2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
8 7 3adant3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
9 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ¬ 𝑝 ≤ 𝑊 )
10 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) )
12 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑝 ∈ 𝐴 )
13 12 9 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) )
14 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑌 ∈ 𝐵 )
15 simp13r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 )
16 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐾 ∈ HL )
17 16 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝐾 ∈ Lat )
18 1 5 atbase ⊢ ( 𝑝 ∈ 𝐴 → 𝑝 ∈ 𝐵 )
19 18 3ad2ant2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑝 ∈ 𝐵 )
20 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑋 ∈ 𝐵 )
21 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑊 ∈ 𝐻 )
22 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
23 21 22 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑊 ∈ 𝐵 )
24 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
25 17 20 23 24 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
26 1 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑝 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → 𝑝 ≤ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) )
27 17 19 25 26 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑝 ≤ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) )
28 simp3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
29 27 28 breqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → 𝑝 ≤ 𝑋 )
30 1 2 3 4 5 6 lhpmcvr4N ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊 ) ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋 ) ) → ¬ 𝑝 ≤ 𝑌 )
31 10 11 13 14 15 29 30 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ¬ 𝑝 ≤ 𝑌 )
32 9 31 28 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ∧ ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) → ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )
33 32 3expia ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) ∧ 𝑝 ∈ 𝐴 ) → ( ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) )
34 33 reximdva ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ( ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ) )
35 8 34 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ∧ ( 𝑌 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ) → ∃ 𝑝 ∈ 𝐴 ( ¬ 𝑝 ≤ 𝑊 ∧ ¬ 𝑝 ≤ 𝑌 ∧ ( 𝑝 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) )