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Metamath Proof Explorer

Theorem cdlemg6c

Description: TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013)

Ref Expression
Hypotheses cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
Assertion cdlemg6c ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) )

Proof

Step Hyp Ref Expression
1 cdlemg4.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemg4.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
3 cdlemg4.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4 cdlemg4.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5 cdlemg4.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
6 cdlemg4.j ⊢ ∨ = ( join ‘ 𝐾 )
7 cdlemg4b.v ⊢ 𝑉 = ( 𝑅 ‘ 𝐺 )
8 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simprl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) )
10 simpl22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
11 simpl23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝐹 ∈ 𝑇 )
12 simpl31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝐺 ∈ 𝑇 )
13 simprr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) )
14 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝐾 ∈ HL )
15 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑄 ∈ 𝐴 )
16 15 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑄 ∈ 𝐴 )
17 simprll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑟 ∈ 𝐴 )
18 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
19 18 3 4 5 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
20 8 12 19 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑅 ‘ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
21 7 20 eqeltrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑉 ∈ ( Base ‘ 𝐾 ) )
22 simp22r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ¬ 𝑄 ≤ 𝑊 )
23 22 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ¬ 𝑄 ≤ 𝑊 )
24 1 3 4 5 trlle ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐺 ) ≤ 𝑊 )
25 8 12 24 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ 𝑊 )
26 7 25 eqbrtrid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑉 ≤ 𝑊 )
27 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐾 ∈ HL )
28 27 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝐾 ∈ Lat )
29 28 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝐾 ∈ Lat )
30 18 2 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
31 15 30 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
32 31 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
33 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑊 ∈ 𝐻 )
34 18 3 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
35 33 34 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
36 35 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
37 18 1 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊 ) → 𝑄 ≤ 𝑊 ) )
38 29 32 21 36 37 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( ( 𝑄 ≤ 𝑉 ∧ 𝑉 ≤ 𝑊 ) → 𝑄 ≤ 𝑊 ) )
39 26 38 mpan2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑄 ≤ 𝑉 → 𝑄 ≤ 𝑊 ) )
40 23 39 mtod ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ¬ 𝑄 ≤ 𝑉 )
41 18 1 6 2 hlexch2 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑄 ≤ 𝑉 ) → ( 𝑄 ≤ ( 𝑟 ∨ 𝑉 ) → 𝑟 ≤ ( 𝑄 ∨ 𝑉 ) ) )
42 14 16 17 21 40 41 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑄 ≤ ( 𝑟 ∨ 𝑉 ) → 𝑟 ≤ ( 𝑄 ∨ 𝑉 ) ) )
43 simpl32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) )
44 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → 𝑃 ∈ 𝐴 )
45 44 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑃 ∈ 𝐴 )
46 18 2 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
47 45 46 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
48 18 1 6 latlej2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ) → 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) )
49 29 47 21 48 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) )
50 18 6 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
51 29 47 21 50 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
52 18 1 6 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) ) ↔ ( 𝑄 ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
53 29 32 21 51 52 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( ( 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ 𝑉 ≤ ( 𝑃 ∨ 𝑉 ) ) ↔ ( 𝑄 ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) ) )
54 43 49 53 mpbi2and ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑄 ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) )
55 18 2 atbase ⊢ ( 𝑟 ∈ 𝐴 → 𝑟 ∈ ( Base ‘ 𝐾 ) )
56 17 55 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → 𝑟 ∈ ( Base ‘ 𝐾 ) )
57 18 6 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ 𝑉 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
58 29 32 21 57 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑄 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) )
59 18 1 lattr ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑟 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑉 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑟 ≤ ( 𝑄 ∨ 𝑉 ) ∧ ( 𝑄 ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) )
60 29 56 58 51 59 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( ( 𝑟 ≤ ( 𝑄 ∨ 𝑉 ) ∧ ( 𝑄 ∨ 𝑉 ) ≤ ( 𝑃 ∨ 𝑉 ) ) → 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) )
61 54 60 mpan2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑟 ≤ ( 𝑄 ∨ 𝑉 ) → 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) )
62 42 61 syld ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑄 ≤ ( 𝑟 ∨ 𝑉 ) → 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) )
63 13 62 mtod ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ¬ 𝑄 ≤ ( 𝑟 ∨ 𝑉 ) )
64 simpl21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
65 simpl33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 )
66 1 2 3 4 5 6 7 cdlemg6a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) = 𝑟 )
67 8 64 9 11 12 13 65 66 syl133anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) = 𝑟 )
68 1 2 3 4 5 6 7 cdlemg6b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑟 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑟 ) ) = 𝑟 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 )
69 8 9 10 11 12 63 67 68 syl133anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 )
70 69 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ∨ 𝑉 ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( ( ( 𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊 ) ∧ ¬ 𝑟 ≤ ( 𝑃 ∨ 𝑉 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) )