Metamath Proof Explorer


Theorem cdlemg11a

Description: TODO: FIX COMMENT. (Contributed by NM, 4-May-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 cdlemg11a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 )

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 simp33 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) )
8 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 )
9 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simpl2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
11 simpl31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → 𝐹𝑇 )
12 simpl32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → 𝐺𝑇 )
13 1 4 5 6 cdlemg6 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺𝑄 ) ) = 𝑄 )
14 9 10 11 12 8 13 syl113anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( 𝐹 ‘ ( 𝐺𝑄 ) ) = 𝑄 )
15 8 14 oveq12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) )
16 15 ex ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) = 𝑃 → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) = ( 𝑃 𝑄 ) ) )
17 16 necon3d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 ) )
18 7 17 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺𝑃 ) ) ≠ 𝑃 )