Metamath Proof Explorer


Theorem cdlemg15a

Description: Eliminate the ( FP ) =/= P condition from cdlemg13 . TODO: FIX COMMENT. (Contributed by NM, 6-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 cdlemg15a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 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 simpl11 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 simpl12 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
10 simpl13 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
11 simpl2l ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → 𝐹𝑇 )
12 simpl2r ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → 𝐺𝑇 )
13 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝐹𝑃 ) = 𝑃 )
14 1 2 3 4 5 6 7 cdlemg14f ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
15 8 9 10 11 12 13 14 syl123anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
16 simpl1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
17 simpl2 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) → ( 𝐹𝑇𝐺𝑇 ) )
18 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) → ( 𝐹𝑃 ) ≠ 𝑃 )
19 simpl3l ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) → ( 𝑅𝐹 ) = ( 𝑅𝐺 ) )
20 simpl3r ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) )
21 1 2 3 4 5 6 7 cdlemg13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝐹𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
22 16 17 18 19 20 21 syl113anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
23 15 22 pm2.61dane ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ∧ ( ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )