Metamath Proof Explorer

Theorem cdlemg13a

Description: TODO: FIX COMMENT. (Contributed by NM, 6-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 cdlemg13a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 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 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL )
9 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
10 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐺 ∈ 𝑇 )
12 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 )
13 10 11 9 12 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 )
14 1 2 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
15 8 9 13 14 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
16 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) )
17 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐹 ∈ 𝑇 )
18 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
19 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
20 10 11 18 19 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) )
21 1 2 3 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) )
22 10 17 20 21 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) )
23 1 2 3 4 5 6 7 trlval2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) )
24 10 11 18 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐺 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) )
25 16 22 24 3eqtr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) )
26 25 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) )
27 1 4 5 6 ltrncoat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 )
28 10 17 11 9 27 syl121anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 )
29 eqid ⊢ ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) = ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 )
30 1 2 3 4 5 29 cdleme0cp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
31 10 20 28 30 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
32 eqid ⊢ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) = ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 )
33 1 2 3 4 5 32 cdleme0cq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐺 ‘ 𝑃 ) ≤ 𝑊 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
34 10 9 20 33 syl12anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ 𝑊 ) ) = ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) )
35 26 31 34 3eqtr3rd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
36 15 35 breqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
37 1 2 4 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
38 8 13 28 37 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
39 8 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ Lat )
40 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
41 40 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
42 9 41 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
43 40 4 atbase ⊢ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
44 28 43 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) )
45 40 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) )
46 8 13 28 45 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) )
47 40 1 2 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) ↔ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) )
48 39 42 44 46 47 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) ↔ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) )
49 36 38 48 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )
50 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
51 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) )
52 1 2 3 4 5 6 cdlemg11a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 )
53 10 18 50 17 11 51 52 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 )
54 53 necomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≠ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) )
55 1 2 4 ps-1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ∧ 𝑃 ≠ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ↔ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) )
56 8 9 28 54 13 28 55 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ≤ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ↔ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) )
57 49 56 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ∧ ( 𝑅 ‘ 𝐹 ) = ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) )