Metamath Proof Explorer


Theorem cdlemg12b

Description: The triples <. P , ( FP ) , ( F( GP ) ) >. and <. Q , ( FQ ) , ( F( GQ ) ) >. are centrally perspective. 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 cdlemg12b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 𝑄 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) )

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 ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 simp2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) )
10 simp31 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐺𝑇 )
11 simp32 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝑃𝑄 )
12 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
13 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝑄𝐴 )
14 simp33 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) )
15 1 2 3 4 5 6 7 cdlemg11b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃 𝑄 ) ≠ ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) )
16 8 12 13 10 11 14 15 syl123anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃 𝑄 ) ≠ ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) )
17 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
18 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝑊𝐻 )
19 eqid ( ( 𝑃 𝑄 ) 𝑊 ) = ( ( 𝑃 𝑄 ) 𝑊 )
20 1 2 3 4 5 19 cdlemg3a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) → ( 𝑃 𝑄 ) = ( 𝑃 ( ( 𝑃 𝑄 ) 𝑊 ) ) )
21 17 18 12 13 20 syl211anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃 𝑄 ) = ( 𝑃 ( ( 𝑃 𝑄 ) 𝑊 ) ) )
22 simp22 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
23 5 6 1 2 4 3 19 cdlemg2k ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝐺𝑇 ) → ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) = ( ( 𝐺𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
24 8 12 22 10 23 syl121anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) = ( ( 𝐺𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
25 16 21 24 3netr3d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃 ( ( 𝑃 𝑄 ) 𝑊 ) ) ≠ ( ( 𝐺𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
26 1 2 3 4 5 6 19 cdlemg12a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ( 𝑃 ( ( 𝑃 𝑄 ) 𝑊 ) ) ≠ ( ( 𝐺𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) ) ) → ( ( 𝑃 ( ( 𝑃 𝑄 ) 𝑊 ) ) ( ( 𝐺𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) ) ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
27 8 9 10 11 25 26 syl113anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( ( 𝑃 𝑄 ) 𝑊 ) ) ( ( 𝐺𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) ) ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
28 21 24 oveq12d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 𝑄 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) = ( ( 𝑃 ( ( 𝑃 𝑄 ) 𝑊 ) ) ( ( 𝐺𝑃 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) ) )
29 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐹𝑇 )
30 5 6 1 2 4 3 19 cdlemg2l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
31 8 12 22 29 10 30 syl122anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
32 27 28 31 3brtr4d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 𝑄 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) )