Metamath Proof Explorer

Theorem cdlemg19a

Description: Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013)

Ref Expression
Hypotheses cdlemg12.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg12.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg12.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemg12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg12.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg12b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg19a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg12.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemg12.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemg12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg12.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg12b.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝐾 ∈ HL )
9 8 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝐾 ∈ Lat )
10 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑃 ∈ 𝐴 )
11 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) )
13 1 4 5 6 ltrncoat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 )
14 11 12 10 13 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 )
15 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 15 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) )
17 8 10 14 16 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) )
18 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑄 ∈ 𝐴 )
19 1 4 5 6 ltrncoat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 )
20 11 12 18 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 )
21 15 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) )
22 8 18 20 21 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) )
23 15 1 3 latmle1 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
24 9 17 22 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
25 1 2 3 4 5 6 7 cdlemg18 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ 𝑊 )
26 1 2 3 4 5 6 7 cdlemg18d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ 𝐴 )
27 15 4 atbase ⊢ ( ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ 𝐴 → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ ( Base ‘ 𝐾 ) )
28 26 27 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ ( Base ‘ 𝐾 ) )
29 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑊 ∈ 𝐻 )
30 15 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
31 29 30 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
32 15 1 3 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ 𝑊 ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) )
33 9 28 17 31 32 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ 𝑊 ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) )
34 24 25 33 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) )
35 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
36 8 35 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝐾 ∈ AtLat )
37 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
38 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
39 simp21l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝐹 ∈ 𝑇 )
40 simp21r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝐺 ∈ 𝑇 )
41 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) )
42 1 2 3 4 5 6 cdlemg11a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 )
43 11 37 38 39 40 41 42 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 )
44 43 necomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → 𝑃 ≠ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )
45 1 2 3 4 5 lhpat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ∧ 𝑃 ≠ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ∈ 𝐴 )
46 11 37 14 44 45 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ∈ 𝐴 )
47 1 4 atcmp ⊢ ( ( 𝐾 ∈ AtLat ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ 𝐴 ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ∈ 𝐴 ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) )
48 36 26 46 47 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ↔ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) )
49 34 48 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) )