Metamath Proof Explorer


Theorem cdlemg17pq

Description: Utility theorem for swapping P and Q . TODO: fix comment. (Contributed by NM, 11-May-2013)

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 cdlemg17pq ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) )

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 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
10 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
11 8 9 10 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) )
12 simp21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐹𝑇 )
13 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐺𝑇 )
14 simp23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑃𝑄 )
15 14 necomd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑄𝑃 )
16 12 13 15 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) )
17 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺𝑃 ) ≠ 𝑃 )
18 1 4 5 6 ltrnatneq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐺𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐺𝑃 ) ≠ 𝑃 ) → ( 𝐺𝑄 ) ≠ 𝑄 )
19 8 13 10 9 17 18 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝐺𝑄 ) ≠ 𝑄 )
20 simp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑅𝐺 ) ( 𝑃 𝑄 ) )
21 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝐾 ∈ HL )
22 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑃𝐴 )
23 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → 𝑄𝐴 )
24 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
25 21 22 23 24 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑃 𝑄 ) = ( 𝑄 𝑃 ) )
26 20 25 breqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( 𝑅𝐺 ) ( 𝑄 𝑃 ) )
27 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) )
28 eqcom ( ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ↔ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) )
29 28 anbi2i ( ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ↔ ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) )
30 29 rexbii ( ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ↔ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) )
31 27 30 sylnib ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) )
32 19 26 31 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) )
33 11 16 32 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑃𝑄 ) ∧ ( ( 𝐺𝑃 ) ≠ 𝑃 ∧ ( 𝑅𝐺 ) ( 𝑃 𝑄 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑃 𝑟 ) = ( 𝑄 𝑟 ) ) ) ) → ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) ∧ ( 𝐹𝑇𝐺𝑇𝑄𝑃 ) ∧ ( ( 𝐺𝑄 ) ≠ 𝑄 ∧ ( 𝑅𝐺 ) ( 𝑄 𝑃 ) ∧ ¬ ∃ 𝑟𝐴 ( ¬ 𝑟 𝑊 ∧ ( 𝑄 𝑟 ) = ( 𝑃 𝑟 ) ) ) ) )