Metamath Proof Explorer


Theorem cdlemg10c

Description: TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in trl* area? (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 )
cdlemg10.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemg10c ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑅𝐹 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ↔ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) )

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 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 simp3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝐹𝑇 )
10 1 5 6 7 trlle ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ) → ( 𝑅𝐹 ) 𝑊 )
11 8 9 10 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝑅𝐹 ) 𝑊 )
12 11 biantrud ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑅𝐹 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ↔ ( ( 𝑅𝐹 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ∧ ( 𝑅𝐹 ) 𝑊 ) ) )
13 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝐾 ∈ HL )
14 13 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝐾 ∈ Lat )
15 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
16 15 5 6 7 trlcl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ) → ( 𝑅𝐹 ) ∈ ( Base ‘ 𝐾 ) )
17 8 9 16 syl2anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝑅𝐹 ) ∈ ( Base ‘ 𝐾 ) )
18 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝐺𝑇 )
19 simp2ll ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝑃𝐴 )
20 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇𝑃𝐴 ) → ( 𝐺𝑃 ) ∈ 𝐴 )
21 8 18 19 20 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝐺𝑃 ) ∈ 𝐴 )
22 simp2rl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝑄𝐴 )
23 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐺𝑇𝑄𝐴 ) → ( 𝐺𝑄 ) ∈ 𝐴 )
24 8 18 22 23 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝐺𝑄 ) ∈ 𝐴 )
25 15 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ ( 𝐺𝑃 ) ∈ 𝐴 ∧ ( 𝐺𝑄 ) ∈ 𝐴 ) → ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
26 13 21 24 25 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
27 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝑊𝐻 )
28 15 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
29 27 28 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
30 15 1 3 latlem12 ( ( 𝐾 ∈ Lat ∧ ( ( 𝑅𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑅𝐹 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ∧ ( 𝑅𝐹 ) 𝑊 ) ↔ ( 𝑅𝐹 ) ( ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) 𝑊 ) ) )
31 14 17 26 29 30 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( ( 𝑅𝐹 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ∧ ( 𝑅𝐹 ) 𝑊 ) ↔ ( 𝑅𝐹 ) ( ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) 𝑊 ) ) )
32 15 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
33 13 19 22 32 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
34 15 1 3 latlem12 ( ( 𝐾 ∈ Lat ∧ ( ( 𝑅𝐹 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ( 𝑅𝐹 ) 𝑊 ) ↔ ( 𝑅𝐹 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
35 14 17 33 29 34 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ( 𝑅𝐹 ) 𝑊 ) ↔ ( 𝑅𝐹 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
36 11 biantrud ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ↔ ( ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ∧ ( 𝑅𝐹 ) 𝑊 ) ) )
37 1 2 3 4 5 6 cdlemg10b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝐺𝑇 ) → ( ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) 𝑊 ) = ( ( 𝑃 𝑄 ) 𝑊 ) )
38 37 3adant3l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) 𝑊 ) = ( ( 𝑃 𝑄 ) 𝑊 ) )
39 38 breq2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑅𝐹 ) ( ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) 𝑊 ) ↔ ( 𝑅𝐹 ) ( ( 𝑃 𝑄 ) 𝑊 ) ) )
40 35 36 39 3bitr4rd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑅𝐹 ) ( ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) 𝑊 ) ↔ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) )
41 12 31 40 3bitrd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇 ) ) → ( ( 𝑅𝐹 ) ( ( 𝐺𝑃 ) ( 𝐺𝑄 ) ) ↔ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) )