Metamath Proof Explorer

Theorem cdleme38m

Description: Part of proof of Lemma E in Crawley p. 113. Show that f(x) is one-to-one on P .\/ Q line. TODO: FIX COMMENT. (Contributed by NM, 13-Mar-2013)

Ref Expression
Hypotheses cdleme38.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme38.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme38.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme38.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme38.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme38.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme38.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme38.d ⊢ 𝐷 = ( ( 𝑢 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) )
cdleme38.v ⊢ 𝑉 = ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 )
cdleme38.x ⊢ 𝑋 = ( ( 𝑢 ∨ 𝐷 ) ∧ 𝑊 )
cdleme38.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑅 ) ∧ 𝑊 ) ) )
cdleme38.g ⊢ 𝐺 = ( ( 𝑆 ∨ 𝑋 ) ∧ ( 𝐷 ∨ ( ( 𝑢 ∨ 𝑆 ) ∧ 𝑊 ) ) )
Assertion cdleme38m ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 = 𝑆 )

Proof

Step Hyp Ref Expression
1 cdleme38.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme38.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme38.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme38.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme38.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme38.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme38.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
8 cdleme38.d ⊢ 𝐷 = ( ( 𝑢 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑢 ) ∧ 𝑊 ) ) )
9 cdleme38.v ⊢ 𝑉 = ( ( 𝑡 ∨ 𝐸 ) ∧ 𝑊 )
10 cdleme38.x ⊢ 𝑋 = ( ( 𝑢 ∨ 𝐷 ) ∧ 𝑊 )
11 cdleme38.f ⊢ 𝐹 = ( ( 𝑅 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑅 ) ∧ 𝑊 ) ) )
12 cdleme38.g ⊢ 𝐺 = ( ( 𝑆 ∨ 𝑋 ) ∧ ( 𝐷 ∨ ( ( 𝑢 ∨ 𝑆 ) ∧ 𝑊 ) ) )
13 simp1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
14 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) )
15 simp311 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
16 simp312 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
17 simp313 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐹 = 𝐺 )
18 15 16 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) )
19 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
20 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) )
21 eqid ⊢ ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑆 ) ∧ 𝑊 ) ) )
22 1 2 3 4 5 6 7 8 9 10 21 12 cdleme37m ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = 𝐺 )
23 13 14 18 19 20 22 syl113anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑆 ) ∧ 𝑊 ) ) ) = 𝐺 )
24 17 23 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝐹 = ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑆 ) ∧ 𝑊 ) ) ) )
25 15 16 24 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) )
26 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
27 26 1 2 3 4 5 6 7 9 11 21 cdleme36m ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = ( ( 𝑆 ∨ 𝑉 ) ∧ ( 𝐸 ∨ ( ( 𝑡 ∨ 𝑆 ) ∧ 𝑊 ) ) ) ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 = 𝑆 )
28 13 14 25 19 27 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝐹 = 𝐺 ) ∧ ( ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( 𝑢 ∈ 𝐴 ∧ ¬ 𝑢 ≤ 𝑊 ) ∧ ¬ 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑅 = 𝑆 )