Metamath Proof Explorer

Theorem cdlemd4

Description: Part of proof of Lemma D in Crawley p. 113. (Contributed by NM, 30-May-2012)

Ref Expression
Hypotheses cdlemd4.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemd4.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemd4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemd4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemd4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemd4 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )

Proof

Step Hyp Ref Expression
1 cdlemd4.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemd4.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemd4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4 cdlemd4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5 cdlemd4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝐾 ∈ HL )
7 simp11r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝑊 ∈ 𝐻 )
8 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
9 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
10 simp231 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ≠ 𝑄 )
11 1 2 3 4 cdlemb2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑠 ∈ 𝐴 ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) )
12 6 7 8 9 10 11 syl221anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ∃ 𝑠 ∈ 𝐴 ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) )
13 simpl11 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simpl12 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) )
15 simpl13 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 ∈ 𝐴 )
16 simpl21 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
17 simprl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑠 ∈ 𝐴 )
18 simprrl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑠 ≤ 𝑊 )
19 17 18 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
20 6 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝐾 ∈ Lat )
21 20 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐾 ∈ Lat )
22 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
23 22 3 atbase ⊢ ( 𝑠 ∈ 𝐴 → 𝑠 ∈ ( Base ‘ 𝐾 ) )
24 23 ad2antrl ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑠 ∈ ( Base ‘ 𝐾 ) )
25 simp21l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
26 22 3 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
27 25 26 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
28 27 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
29 simp22l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
30 22 3 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
31 29 30 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
32 31 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
33 simprrr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) )
34 22 1 2 latnlej1l ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑠 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≠ 𝑃 )
35 34 necomd ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑠 ∈ ( Base ‘ 𝐾 ) ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑠 )
36 21 24 28 32 33 35 syl131anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ≠ 𝑠 )
37 simpl22 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
38 simpl23 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) )
39 1 2 3 4 cdlemd3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑠 ) )
40 13 16 37 38 15 17 33 39 syl133anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑠 ) )
41 36 40 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ≠ 𝑠 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑠 ) ) )
42 simpl3l ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
43 10 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ≠ 𝑄 )
44 43 33 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) )
45 simpl3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) )
46 1 2 3 4 5 cdlemd2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑠 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑠 ) = ( 𝐺 ‘ 𝑠 ) )
47 13 14 17 16 37 44 45 46 syl331anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐹 ‘ 𝑠 ) = ( 𝐺 ‘ 𝑠 ) )
48 1 2 3 4 5 cdlemd2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑠 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑠 ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑠 ) = ( 𝐺 ‘ 𝑠 ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )
49 13 14 15 16 19 41 42 47 48 syl332anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )
50 12 49 rexlimddv ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )