Metamath Proof Explorer

Theorem lhp2at0ne

Description: Inequality for joins with 2 different atoms (or an atom and zero) under co-atom W . (Contributed by NM, 28-Jul-2013)

Ref Expression
Hypotheses lhp2at0nle.l ⊢ ≤ = ( le ‘ 𝐾 )
lhp2at0nle.j ⊢ ∨ = ( join ‘ 𝐾 )
lhp2at0nle.z ⊢ 0 = ( 0. ‘ 𝐾 )
lhp2at0nle.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
lhp2at0nle.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion lhp2at0ne ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( 𝑃 ∨ 𝑈 ) ≠ ( 𝑄 ∨ 𝑉 ) )

Proof

Step Hyp Ref Expression
1 lhp2at0nle.l ⊢ ≤ = ( le ‘ 𝐾 )
2 lhp2at0nle.j ⊢ ∨ = ( join ‘ 𝐾 )
3 lhp2at0nle.z ⊢ 0 = ( 0. ‘ 𝐾 )
4 lhp2at0nle.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 lhp2at0nle.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
7 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
8 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → 𝑈 ≠ 𝑉 )
9 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) )
10 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
11 1 2 3 4 5 lhp2at0nle ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑈 ≠ 𝑉 ) ∧ ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) → ¬ 𝑉 ≤ ( 𝑃 ∨ 𝑈 ) )
12 6 7 8 9 10 11 syl311anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ¬ 𝑉 ≤ ( 𝑃 ∨ 𝑈 ) )
13 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → 𝐾 ∈ HL )
14 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → 𝑄 ∈ 𝐴 )
15 simp2rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → 𝑉 ∈ 𝐴 )
16 1 2 4 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ) → 𝑉 ≤ ( 𝑄 ∨ 𝑉 ) )
17 13 14 15 16 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → 𝑉 ≤ ( 𝑄 ∨ 𝑉 ) )
18 17 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) ∧ ( 𝑃 ∨ 𝑈 ) = ( 𝑄 ∨ 𝑉 ) ) → 𝑉 ≤ ( 𝑄 ∨ 𝑉 ) )
19 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) ∧ ( 𝑃 ∨ 𝑈 ) = ( 𝑄 ∨ 𝑉 ) ) → ( 𝑃 ∨ 𝑈 ) = ( 𝑄 ∨ 𝑉 ) )
20 18 19 breqtrrd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) ∧ ( 𝑃 ∨ 𝑈 ) = ( 𝑄 ∨ 𝑉 ) ) → 𝑉 ≤ ( 𝑃 ∨ 𝑈 ) )
21 20 ex ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( ( 𝑃 ∨ 𝑈 ) = ( 𝑄 ∨ 𝑉 ) → 𝑉 ≤ ( 𝑃 ∨ 𝑈 ) ) )
22 21 necon3bd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( ¬ 𝑉 ≤ ( 𝑃 ∨ 𝑈 ) → ( 𝑃 ∨ 𝑈 ) ≠ ( 𝑄 ∨ 𝑉 ) ) )
23 12 22 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( ( ( 𝑈 ∈ 𝐴 ∨ 𝑈 = 0 ) ∧ 𝑈 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ 𝑈 ≠ 𝑉 ) → ( 𝑃 ∨ 𝑈 ) ≠ ( 𝑄 ∨ 𝑉 ) )