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