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Metamath Proof Explorer

Theorem cdleme21d

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 115, 3rd line. D , F , N , E , B , Z represent s_2, f(s), f_s(r), z_2, f(z), f_z(r) respectively. We prove f_s(r) = f_z(r). (Contributed by NM, 29-Nov-2012)

Ref Expression
Hypotheses cdleme21.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme21.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme21.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme21.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme21.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme21.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme21.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme21.b ⊢ 𝐵 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme21.d ⊢ 𝐷 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
cdleme21.e ⊢ 𝐸 = ( ( 𝑅 ∨ 𝑧 ) ∧ 𝑊 )
cdleme21d.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐷 ) )
cdleme21d.z ⊢ 𝑍 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐵 ∨ 𝐸 ) )
Assertion cdleme21d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑁 = 𝑍 )

Proof

Step Hyp Ref Expression
1 cdleme21.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme21.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme21.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme21.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme21.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme21.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme21.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8 cdleme21.b ⊢ 𝐵 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
9 cdleme21.d ⊢ 𝐷 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
10 cdleme21.e ⊢ 𝐸 = ( ( 𝑅 ∨ 𝑧 ) ∧ 𝑊 )
11 cdleme21d.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ 𝐷 ) )
12 cdleme21d.z ⊢ 𝑍 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐵 ∨ 𝐸 ) )
13 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
15 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
16 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
17 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
18 simp33l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) )
19 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑃 ≠ 𝑄 )
20 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝐾 ∈ HL )
21 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑃 ∈ 𝐴 )
22 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑄 ∈ 𝐴 )
23 simp2rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑆 ∈ 𝐴 )
24 simp32l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
25 18 simpld ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑧 ∈ 𝐴 )
26 simp33r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) )
27 1 2 4 cdleme21a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → 𝑆 ≠ 𝑧 )
28 20 21 22 23 24 25 26 27 syl322anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑆 ≠ 𝑧 )
29 19 28 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑧 ) )
30 1 2 4 cdleme21b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) )
31 20 21 22 23 19 24 25 26 30 syl332anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) )
32 simp32r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
33 24 31 32 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
34 1 2 3 4 5 6 cdleme21c ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) → ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) )
35 13 14 22 23 19 24 25 26 34 syl332anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) )
36 eqid ⊢ ( ( 𝑆 ∨ 𝑧 ) ∧ 𝑊 ) = ( ( 𝑆 ∨ 𝑧 ) ∧ 𝑊 )
37 1 2 3 4 5 6 7 8 9 10 36 11 12 cdleme20 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑧 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑧 ) ) ) → 𝑁 = 𝑍 )
38 13 14 15 16 17 18 29 33 35 37 syl333anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) ) → 𝑁 = 𝑍 )