Metamath Proof Explorer

Theorem dihmeetlem3N

Description: Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dihmeetlem3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihmeetlem3.l ⊢ ≤ = ( le ‘ 𝐾 )
dihmeetlem3.j ⊢ ∨ = ( join ‘ 𝐾 )
dihmeetlem3.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihmeetlem3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihmeetlem3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion dihmeetlem3N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑄 ≠ 𝑅 )

Proof

Step Hyp Ref Expression
1 dihmeetlem3.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihmeetlem3.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihmeetlem3.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihmeetlem3.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihmeetlem3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 dihmeetlem3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 simp2lr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ¬ 𝑄 ≤ 𝑊 )
8 oveq1 ⊢ ( 𝑄 = 𝑅 → ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) )
9 simpr ⊢ ( ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 )
10 8 9 sylan9eqr ⊢ ( ( ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ∧ 𝑄 = 𝑅 ) → ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 )
11 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ HL )
12 11 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝐾 ∈ Lat )
13 simp2ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ∈ 𝐴 )
14 1 5 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ 𝐵 )
15 13 14 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ∈ 𝐵 )
16 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑋 ∈ 𝐵 )
17 simp12r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑌 ∈ 𝐵 )
18 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
19 12 16 17 18 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )
20 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑊 ∈ 𝐻 )
21 1 6 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ 𝐵 )
22 20 21 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑊 ∈ 𝐵 )
23 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
24 12 16 22 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 )
25 1 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ ( 𝑋 ∧ 𝑊 ) ∈ 𝐵 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) )
26 12 15 24 25 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) )
27 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 )
28 26 27 breqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ≤ 𝑋 )
29 1 4 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
30 12 17 22 29 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 )
31 1 2 3 latlej1 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐵 ∧ ( 𝑌 ∧ 𝑊 ) ∈ 𝐵 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) )
32 12 15 30 31 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) )
33 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 )
34 32 33 breqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ≤ 𝑌 )
35 1 2 4 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝑄 ≤ 𝑋 ∧ 𝑄 ≤ 𝑌 ) ↔ 𝑄 ≤ ( 𝑋 ∧ 𝑌 ) ) )
36 12 15 16 17 35 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( ( 𝑄 ≤ 𝑋 ∧ 𝑄 ≤ 𝑌 ) ↔ 𝑄 ≤ ( 𝑋 ∧ 𝑌 ) ) )
37 28 34 36 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ≤ ( 𝑋 ∧ 𝑌 ) )
38 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 )
39 1 2 12 15 19 22 37 38 lattrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → 𝑄 ≤ 𝑊 )
40 39 3exp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) → ( ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ( 𝑄 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 → 𝑄 ≤ 𝑊 ) ) )
41 10 40 syl7 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) → ( ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ( ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ∧ 𝑄 = 𝑅 ) → 𝑄 ≤ 𝑊 ) ) )
42 41 exp4a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) → ( ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → ( ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) → ( 𝑄 = 𝑅 → 𝑄 ≤ 𝑊 ) ) ) )
43 42 3imp ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( 𝑄 = 𝑅 → 𝑄 ≤ 𝑊 ) )
44 43 necon3bd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → ( ¬ 𝑄 ≤ 𝑊 → 𝑄 ≠ 𝑅 ) )
45 7 44 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑊 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑄 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑅 ∨ ( 𝑌 ∧ 𝑊 ) ) = 𝑌 ) ) → 𝑄 ≠ 𝑅 )