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Metamath Proof Explorer

Theorem cdlemg18d

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 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 )

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 simp1 ⊢ ( ( ( ( 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 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
9 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 )
10 simp22 ⊢ ( ( ( ( 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 ≠ 𝑄 )
11 simp23 ⊢ ( ( ( ( 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 ‘ 𝑃 ) ≠ 𝑃 )
12 simp31 ⊢ ( ( ( ( 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 ) ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) )
13 simp33 ⊢ ( ( ( ( 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 ) ) ) ) → ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) )
14 1 2 3 4 5 6 7 cdlemg17b ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐺 ‘ 𝑃 ) = 𝑄 )
15 8 9 10 11 12 13 14 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 ) ) ) ) → ( 𝐺 ‘ 𝑃 ) = 𝑄 )
16 15 fveq2d ⊢ ( ( ( ( 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 ‘ 𝑄 ) )
17 16 oveq2d ⊢ ( ( ( ( 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 ) ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) )
18 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 )
19 1 2 3 4 5 6 7 cdlemg17bq ⊢ ( ( ( ( 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 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐺 ‘ 𝑄 ) = 𝑃 )
20 8 18 9 10 11 12 13 19 syl133anc ⊢ ( ( ( ( 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 ‘ 𝑄 ) = 𝑃 )
21 20 fveq2d ⊢ ( ( ( ( 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 ) ) = ( 𝐹 ‘ 𝑃 ) )
22 21 oveq2d ⊢ ( ( ( ( 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 ) ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) )
23 17 22 oveq12d ⊢ ( ( ( ( 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 ‘ 𝑄 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) )
24 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 ) )
25 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 ) )
26 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 ) )
27 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 ) )
28 1 2 3 4 5 6 cdlemg11aq ⊢ ( ( ( 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 ‘ 𝑄 ) ) ≠ 𝑄 )
29 24 25 26 18 9 27 28 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 ‘ 𝑄 ) ) ≠ 𝑄 )
30 21 29 eqnetrrd ⊢ ( ( ( ( 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 ‘ 𝑃 ) ≠ 𝑄 )
31 1 2 3 4 5 6 7 cdlemg17irq ⊢ ( ( ( ( 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 ) ∧ ¬ ∃ 𝑟 ∈ 𝐴 ( ¬ 𝑟 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑟 ) = ( 𝑄 ∨ 𝑟 ) ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = ( 𝐹 ‘ 𝑃 ) )
32 8 18 9 10 11 12 13 31 syl133anc ⊢ ( ( ( ( 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 ) ) = ( 𝐹 ‘ 𝑃 ) )
33 16 32 oveq12d ⊢ ( ( ( ( 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 ) ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
34 33 27 eqnetrrd ⊢ ( ( ( ( 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 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) )
35 eqid ⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
36 1 2 3 4 5 6 7 35 cdlemg18c ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∈ 𝐴 )
37 24 25 26 18 10 30 34 36 syl133anc ⊢ ( ( ( ( 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 ‘ 𝑄 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ∈ 𝐴 )
38 23 37 eqeltrd ⊢ ( ( ( ( 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 )