Metamath Proof Explorer

Theorem cdleme35f

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

Ref Expression
Hypotheses cdleme35.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme35.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme35.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme35.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme35.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme35.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme35.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ) )
Assertion cdleme35f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑃 ∨ 𝑅 ) ) = 𝑅 )

Proof

Step Hyp Ref Expression
1 cdleme35.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme35.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme35.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme35.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme35.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme35.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme35.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑅 ) ∧ 𝑊 ) ) )
8 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ HL )
9 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ 𝐴 )
10 simp2rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ∈ 𝐴 )
11 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑃 ) )
12 8 9 10 11 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝑅 ) = ( 𝑅 ∨ 𝑃 ) )
13 12 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑃 ∨ 𝑅 ) ) = ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑃 ) ) )
14 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
16 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ∈ 𝐴 )
17 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 )
18 1 2 3 4 5 6 cdleme0a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑈 ∈ 𝐴 )
19 14 15 16 17 18 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑈 ∈ 𝐴 )
20 simp12r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑃 ≤ 𝑊 )
21 8 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ Lat )
22 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
23 22 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
24 8 9 16 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
25 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑊 ∈ 𝐻 )
26 22 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
27 25 26 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
28 22 1 3 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
29 21 24 27 28 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
30 6 29 eqbrtrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑈 ≤ 𝑊 )
31 breq1 ⊢ ( 𝑈 = 𝑃 → ( 𝑈 ≤ 𝑊 ↔ 𝑃 ≤ 𝑊 ) )
32 30 31 syl5ibcom ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑈 = 𝑃 → 𝑃 ≤ 𝑊 ) )
33 32 necon3bd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ¬ 𝑃 ≤ 𝑊 → 𝑈 ≠ 𝑃 ) )
34 20 33 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑈 ≠ 𝑃 )
35 simp3 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
36 22 1 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
37 21 24 27 36 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ ( 𝑃 ∨ 𝑄 ) )
38 6 37 eqbrtrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) )
39 1 2 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) )
40 8 9 16 39 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) )
41 22 4 atbase ⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
42 19 41 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
43 22 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
44 9 43 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
45 22 1 2 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑈 ∨ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
46 21 42 44 24 45 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑈 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( 𝑈 ∨ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
47 38 40 46 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑈 ∨ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) )
48 22 4 atbase ⊢ ( 𝑅 ∈ 𝐴 → 𝑅 ∈ ( Base ‘ 𝐾 ) )
49 10 48 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
50 22 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑈 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
51 8 19 9 50 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑈 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
52 22 1 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑅 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑈 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑅 ≤ ( 𝑈 ∨ 𝑃 ) ∧ ( 𝑈 ∨ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
53 21 49 51 24 52 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑅 ≤ ( 𝑈 ∨ 𝑃 ) ∧ ( 𝑈 ∨ 𝑃 ) ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
54 47 53 mpan2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑅 ≤ ( 𝑈 ∨ 𝑃 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
55 35 54 mtod ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑅 ≤ ( 𝑈 ∨ 𝑃 ) )
56 1 2 3 4 2llnma2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑈 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑈 ≠ 𝑃 ∧ ¬ 𝑅 ≤ ( 𝑈 ∨ 𝑃 ) ) ) → ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑃 ) ) = 𝑅 )
57 8 19 9 10 34 55 56 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑅 ∨ 𝑃 ) ) = 𝑅 )
58 13 57 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑅 ∨ 𝑈 ) ∧ ( 𝑃 ∨ 𝑅 ) ) = 𝑅 )