Metamath Proof Explorer


Theorem cdlemg8

Description: TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013)

Ref Expression
Hypotheses cdlemg8.l = ( le ‘ 𝐾 )
cdlemg8.j = ( join ‘ 𝐾 )
cdlemg8.m = ( meet ‘ 𝐾 )
cdlemg8.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg8.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg8.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg8 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )

Proof

Step Hyp Ref Expression
1 cdlemg8.l = ( le ‘ 𝐾 )
2 cdlemg8.j = ( join ‘ 𝐾 )
3 cdlemg8.m = ( meet ‘ 𝐾 )
4 cdlemg8.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg8.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg8.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
8 simpl21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
9 simpl22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
10 simpl23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → 𝐹𝑇 )
11 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → 𝐺𝑇 )
12 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 )
13 1 2 3 4 5 6 cdlemg8a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
14 7 8 9 10 11 12 13 syl123anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
15 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
16 simpl2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) → ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) )
17 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) → 𝐺𝑇 )
18 simpl3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) )
19 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 )
20 1 2 3 4 5 6 cdlemg8d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
21 15 16 17 18 19 20 syl113anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
22 14 21 pm2.61dane ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )