Metamath Proof Explorer

Theorem cdlemg18a

Description: Show two lines are different. TODO: fix comment. (Contributed by NM, 14-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 cdlemg18a ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) )

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 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) )
9 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝐾 ∈ HL )
10 simpl21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝑃 ∈ 𝐴 )
11 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simpl23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝐹 ∈ 𝑇 )
13 simpl22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝑄 ∈ 𝐴 )
14 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
15 11 12 13 14 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
16 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
17 11 12 10 16 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
18 simpl3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝑃 ≠ 𝑄 )
19 4 5 6 ltrn11at ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑄 ) )
20 11 12 10 13 18 19 syl113anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( 𝐹 ‘ 𝑄 ) )
21 20 necomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝐹 ‘ 𝑄 ) ≠ ( 𝐹 ‘ 𝑃 ) )
22 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) )
23 2 4 hlatexch4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑄 ) ≠ ( 𝐹 ‘ 𝑃 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
24 9 10 15 13 17 18 21 22 23 syl323anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
25 24 eqcomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) )
26 25 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) ) )
27 26 necon3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) ) )
28 8 27 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑄 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑃 ) ) )