Metamath Proof Explorer


Theorem cdlemg12c

Description: The triples <. P , ( FP ) , ( F( GP ) ) >. and <. Q , ( FQ ) , ( F( GQ ) ) >. are axially perspective by dalaw . TODO: FIX COMMENT. (Contributed by NM, 5-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 cdlemg12c ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) ( 𝑄 ( 𝐺𝑄 ) ) ) ( ( ( ( 𝐺𝑃 ) ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( ( 𝐺𝑄 ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) 𝑃 ) ( ( 𝐹 ‘ ( 𝐺𝑄 ) ) 𝑄 ) ) ) )

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 1 2 3 4 5 6 7 cdlemg12b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 𝑄 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) )
9 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
10 simp21l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝑃𝐴 )
11 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
12 simp31 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐺𝑇 )
13 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇𝑃𝐴 ) → ( 𝐺𝑃 ) ∈ 𝐴 )
14 11 12 10 13 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝐺𝑃 ) ∈ 𝐴 )
15 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐹𝑇 )
16 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝐺𝑃 ) ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) ∈ 𝐴 )
17 11 15 14 16 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) ∈ 𝐴 )
18 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝑄𝐴 )
19 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇𝑄𝐴 ) → ( 𝐺𝑄 ) ∈ 𝐴 )
20 11 12 18 19 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝐺𝑄 ) ∈ 𝐴 )
21 1 4 5 6 ltrncoat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ 𝑄𝐴 ) → ( 𝐹 ‘ ( 𝐺𝑄 ) ) ∈ 𝐴 )
22 11 15 12 18 21 syl121anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺𝑄 ) ) ∈ 𝐴 )
23 1 2 3 4 dalaw ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴 ∧ ( 𝐺𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ∈ 𝐴 ) ∧ ( 𝑄𝐴 ∧ ( 𝐺𝑄 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺𝑄 ) ) ∈ 𝐴 ) ) → ( ( ( 𝑃 𝑄 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) ( 𝑄 ( 𝐺𝑄 ) ) ) ( ( ( ( 𝐺𝑃 ) ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( ( 𝐺𝑄 ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) 𝑃 ) ( ( 𝐹 ‘ ( 𝐺𝑄 ) ) 𝑄 ) ) ) ) )
24 9 10 14 17 18 20 22 23 syl133anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( ( 𝑃 𝑄 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) ( 𝑄 ( 𝐺𝑄 ) ) ) ( ( ( ( 𝐺𝑃 ) ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( ( 𝐺𝑄 ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) 𝑃 ) ( ( 𝐹 ‘ ( 𝐺𝑄 ) ) 𝑄 ) ) ) ) )
25 8 24 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) ( 𝑄 ( 𝐺𝑄 ) ) ) ( ( ( ( 𝐺𝑃 ) ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) ( ( 𝐺𝑄 ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ) ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) 𝑃 ) ( ( 𝐹 ‘ ( 𝐺𝑄 ) ) 𝑄 ) ) ) )