Metamath Proof Explorer


Theorem cdlemg11b

Description: TODO: FIX COMMENT. (Contributed by NM, 5-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 )
cdlemg10.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg11b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃 𝑄 ) ≠ ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) )

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 cdlemg10.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp33 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) )
9 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simpl31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝐺𝑇 )
11 simpl2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
12 1 2 3 4 5 6 7 trlval2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑅𝐺 ) = ( ( 𝑃 ( 𝐺𝑃 ) ) 𝑊 ) )
13 9 10 11 12 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝑅𝐺 ) = ( ( 𝑃 ( 𝐺𝑃 ) ) 𝑊 ) )
14 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
15 simpl1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝐾 ∈ HL )
16 15 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝐾 ∈ Lat )
17 simp2ll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → 𝑃𝐴 )
18 17 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝑃𝐴 )
19 14 4 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
20 18 19 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
21 14 5 6 ltrncl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺𝑃 ) ∈ ( Base ‘ 𝐾 ) )
22 9 10 20 21 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝐺𝑃 ) ∈ ( Base ‘ 𝐾 ) )
23 14 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺𝑃 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ( 𝐺𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
24 16 20 22 23 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝑃 ( 𝐺𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
25 simpl1r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝑊𝐻 )
26 14 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
27 25 26 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
28 14 3 latmcl ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ( 𝐺𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
29 16 24 27 28 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
30 simpl2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝑄𝐴 )
31 14 4 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
32 30 31 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
33 14 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
34 16 20 32 33 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
35 14 1 3 latmle1 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ( 𝐺𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) 𝑊 ) ( 𝑃 ( 𝐺𝑃 ) ) )
36 16 24 27 35 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) 𝑊 ) ( 𝑃 ( 𝐺𝑃 ) ) )
37 14 1 2 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ( 𝑃 𝑄 ) )
38 16 20 32 37 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → 𝑃 ( 𝑃 𝑄 ) )
39 14 5 6 ltrncl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺𝑄 ) ∈ ( Base ‘ 𝐾 ) )
40 9 10 32 39 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝐺𝑄 ) ∈ ( Base ‘ 𝐾 ) )
41 14 1 2 latlej1 ( ( 𝐾 ∈ Lat ∧ ( 𝐺𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝐺𝑃 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) )
42 16 22 40 41 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝐺𝑃 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) )
43 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) )
44 42 43 breqtrrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝐺𝑃 ) ( 𝑃 𝑄 ) )
45 14 1 2 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐺𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ( 𝑃 𝑄 ) ∧ ( 𝐺𝑃 ) ( 𝑃 𝑄 ) ) ↔ ( 𝑃 ( 𝐺𝑃 ) ) ( 𝑃 𝑄 ) ) )
46 16 20 22 34 45 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( ( 𝑃 ( 𝑃 𝑄 ) ∧ ( 𝐺𝑃 ) ( 𝑃 𝑄 ) ) ↔ ( 𝑃 ( 𝐺𝑃 ) ) ( 𝑃 𝑄 ) ) )
47 38 44 46 mpbi2and ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝑃 ( 𝐺𝑃 ) ) ( 𝑃 𝑄 ) )
48 14 1 16 29 24 34 36 47 lattrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( ( 𝑃 ( 𝐺𝑃 ) ) 𝑊 ) ( 𝑃 𝑄 ) )
49 13 48 eqbrtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) → ( 𝑅𝐺 ) ( 𝑃 𝑄 ) )
50 49 ex ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 𝑄 ) = ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) → ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) )
51 50 necon3bd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) → ( 𝑃 𝑄 ) ≠ ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ) )
52 8 51 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ∧ ( 𝐺𝑇𝑃𝑄 ∧ ¬ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃 𝑄 ) ≠ ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) )