Metamath Proof Explorer


Theorem cdlemg16ALTN

Description: This version of cdlemg16 uses cdlemg15a instead of cdlemg15 , in case cdlemg15 ends up not being needed. TODO: FIX COMMENT. (Contributed by NM, 6-May-2013) (New usage is discouraged.)

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 cdlemg16ALTN ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 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 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → 𝐾 ∈ HL )
9 simpl12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → 𝑊𝐻 )
10 8 9 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
11 simpl21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
12 simpl22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
13 simpl13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → ( 𝐹𝑇𝐺𝑇 ) )
14 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → ( 𝑅𝐹 ) = ( 𝑅𝐺 ) )
15 simpl31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) )
16 1 2 3 4 5 6 7 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 ) )
17 10 11 12 13 14 15 16 syl312anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) = ( 𝑅𝐺 ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
18 simpl11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → 𝐾 ∈ HL )
19 simpl12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → 𝑊𝐻 )
20 18 19 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
21 simpl21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
22 simpl22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
23 simp13l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐹𝑇 )
24 23 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → 𝐹𝑇 )
25 simp13r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝐺𝑇 )
26 25 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → 𝐺𝑇 )
27 simpl23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → 𝑃𝑄 )
28 simpl32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) )
29 simpl33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) )
30 28 29 jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ( ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) )
31 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) )
32 simpl31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) )
33 1 2 3 4 5 6 7 cdlemg12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
34 20 21 22 24 26 27 30 31 32 33 syl333anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑅𝐹 ) ≠ ( 𝑅𝐺 ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )
35 17 34 pm2.61dane ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ∧ ( 𝐹𝑇𝐺𝑇 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝑃𝑄 ) ∧ ( ( ( 𝐹 ‘ ( 𝐺𝑃 ) ) ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) ≠ ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹 ‘ ( 𝐺𝑃 ) ) ) 𝑊 ) = ( ( 𝑄 ( 𝐹 ‘ ( 𝐺𝑄 ) ) ) 𝑊 ) )