Metamath Proof Explorer


Theorem cdlemg18

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 cdlemg18 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) 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 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 simp21r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐺𝑇 )
10 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
11 1 2 3 4 5 6 7 cdlemg18d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ∈ 𝐴 )
12 simp23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺𝑃 ) ≠ 𝑃 )
13 simp1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
14 simp21l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐹𝑇 )
15 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑃𝑄 )
16 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑅𝐺 ) ( 𝑃 𝑄 ) )
17 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) )
18 1 2 3 4 5 6 7 cdlemg17 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺 ‘ ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ) = ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) )
19 13 14 9 15 12 16 17 18 syl133anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺 ‘ ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ) = ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) )
20 1 4 5 6 ltrnatlw ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐺𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ∈ 𝐴 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝐺 ‘ ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ) = ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) 𝑊 )
21 8 9 10 11 12 19 20 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑃𝑄 ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) ∧ ( ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) 𝑊 )