Metamath Proof Explorer

Theorem cdlemg10a

Description: TODO: FIX COMMENT. (Contributed by NM, 3-May-2013)

Ref Expression
Hypotheses cdlemg8.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg8.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg8.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg10.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg10a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝑅 ‘ 𝐺 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg8.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg8.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg8.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemg8.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg8.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg8.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg10.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
10 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
11 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐹 ∈ 𝑇 )
12 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐺 ∈ 𝑇 )
13 simp23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≠ 𝑄 )
14 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) )
15 1 2 3 4 5 6 cdlemg9 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ∨ ( ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑄 ) ) ) )
16 8 9 10 11 12 13 14 15 syl133anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ∨ ( ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑄 ) ) ) )
17 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
18 8 12 9 17 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
19 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑄 ) ≤ 𝑊 ) )
20 8 12 10 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑄 ) ≤ 𝑊 ) )
21 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
22 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
23 4 5 6 ltrn11at ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝐺 ‘ 𝑃 ) ≠ ( 𝐺 ‘ 𝑄 ) )
24 8 12 21 22 13 23 syl113anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ≠ ( 𝐺 ‘ 𝑄 ) )
25 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) )
26 1 2 3 4 5 6 7 cdlemg10c ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
27 8 9 10 11 12 26 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ↔ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
28 25 27 mtbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
29 1 2 3 4 5 6 7 trlval4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑄 ) ≤ 𝑊 ) ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ ( 𝐺 ‘ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) )
30 8 11 18 20 24 28 29 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) )
31 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL )
32 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 )
33 8 12 21 32 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 )
34 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 )
35 8 11 33 34 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 )
36 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐺 ‘ 𝑃 ) ) )
37 31 33 35 36 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐺 ‘ 𝑃 ) ) )
38 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 )
39 8 12 22 38 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 )
40 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 )
41 8 11 39 40 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 )
42 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
43 31 39 41 42 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ ( 𝐺 ‘ 𝑄 ) ) )
44 37 43 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
45 30 44 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
46 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) )
47 1 2 3 4 5 6 7 trlval4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
48 8 12 9 10 13 46 47 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) )
49 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑃 ) )
50 31 21 33 49 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑃 ) )
51 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ) → ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) = ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑄 ) )
52 31 22 39 51 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) = ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑄 ) )
53 50 52 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑄 ) ) )
54 48 53 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑄 ) ) )
55 45 54 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝑅 ‘ 𝐺 ) ) = ( ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ∨ ( ( ( 𝐺 ‘ 𝑃 ) ∨ 𝑃 ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ 𝑄 ) ) ) )
56 16 55 breqtrrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑅 ‘ 𝐹 ) ∨ ( 𝑅 ‘ 𝐺 ) ) )