Metamath Proof Explorer

Theorem cdlemd5

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 cdlemd5 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 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 fveq2 ⊢ ( 𝑅 = 𝑃 → ( 𝐹 ‘ 𝑅 ) = ( 𝐹 ‘ 𝑃 ) )
7 fveq2 ⊢ ( 𝑅 = 𝑃 → ( 𝐺 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑃 ) )
8 6 7 eqeq12d ⊢ ( 𝑅 = 𝑃 → ( ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) ↔ ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ) )
9 simpll1 ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) )
10 simpl21 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
11 10 adantr ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
12 simpl22 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
13 12 adantr ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
14 simp23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → 𝑃 ≠ 𝑄 )
15 14 ad2antrr ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → 𝑃 ≠ 𝑄 )
16 simplr ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
17 simpr ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → 𝑅 ≠ 𝑃 )
18 15 16 17 3jca ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑃 ) )
19 simpll3 ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) )
20 1 2 3 4 5 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 ) )
21 9 11 13 18 19 20 syl131anc ⊢ ( ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≠ 𝑃 ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )
22 simpl3l ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) )
23 8 21 22 pm2.61ne ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )
24 simpl1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) )
25 simpl21 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
26 simpl22 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
27 simpl23 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 )
28 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
29 27 28 jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
30 simpl3 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) )
31 1 2 3 4 5 cdlemd2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )
32 24 25 26 29 30 31 syl131anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )
33 23 32 pm2.61dan ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑅 ∈ 𝐴 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) = ( 𝐺 ‘ 𝑃 ) ∧ ( 𝐹 ‘ 𝑄 ) = ( 𝐺 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑅 ) = ( 𝐺 ‘ 𝑅 ) )