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Metamath Proof Explorer

Theorem cdlemg4

Description: TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013)

Ref Expression
Hypotheses cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
Assertion cdlemg4 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 )

Proof

Step Hyp Ref Expression
1 cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
7 cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
8 eqid ⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
9 1 2 3 4 5 6 7 8 cdlemg4g ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = ( ( 𝑄 ∨ 𝑉 ) ( meet ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) )
10 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐾 ∈ HL )
11 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑃 ∈ 𝐴 )
12 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑄 ∈ 𝐴 )
13 6 2 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) )
14 10 11 12 13 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑄 ∨ 𝑃 ) )
15 14 oveq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( 𝑄 ∨ 𝑉 ) ( meet ‘ 𝐾 ) ( 𝑃 ∨ 𝑄 ) ) = ( ( 𝑄 ∨ 𝑉 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑃 ) ) )
16 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17 simp31 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐺 ∈ 𝑇 )
18 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
19 18 3 4 5 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
20 16 17 19 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑅 ‘ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
21 7 20 eqeltrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑉 ∈ ( Base ‘ 𝐾 ) )
22 simp32 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) )
23 simp21r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ¬ 𝑃 ≤ 𝑊 )
24 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
25 1 6 8 2 3 4 5 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
26 16 17 24 25 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
27 7 26 eqtrid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑉 = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) )
28 10 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐾 ∈ Lat )
29 1 2 3 4 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
30 16 17 24 29 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
31 30 simpld ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 )
32 18 6 2 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
33 10 11 31 32 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
34 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑊 ∈ 𝐻 )
35 18 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
36 34 35 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
37 18 1 8 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 )
38 28 33 36 37 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ( meet ‘ 𝐾 ) 𝑊 ) ≤ 𝑊 )
39 27 38 eqbrtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑉 ≤ 𝑊 )
40 18 2 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
41 11 40 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
42 18 1 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊 ) → 𝑃 ≤ 𝑊 ) )
43 28 41 21 36 42 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( 𝑃 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊 ) → 𝑃 ≤ 𝑊 ) )
44 39 43 mpan2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ≤ 𝑉 → 𝑃 ≤ 𝑊 ) )
45 23 44 mtod ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ¬ 𝑃 ≤ 𝑉 )
46 18 1 6 2 hlexch2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑃 ≤ 𝑉 ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑉 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ) )
47 10 11 12 21 45 46 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝑃 ≤ ( 𝑄 ∨ 𝑉 ) → 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ) )
48 22 47 mtod ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑉 ) )
49 18 1 6 8 2 2llnma1b ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ ¬ 𝑃 ≤ ( 𝑄 ∨ 𝑉 ) ) → ( ( 𝑄 ∨ 𝑉 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑃 ) ) = 𝑄 )
50 10 21 12 11 48 49 syl131anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( 𝑄 ∨ 𝑉 ) ( meet ‘ 𝐾 ) ( 𝑄 ∨ 𝑃 ) ) = 𝑄 )
51 9 15 50 3eqtrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 )