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Metamath Proof Explorer

Theorem cdleme40n

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. TODO get rid of '.<' class? (Contributed by NM, 18-Mar-2013)

Ref Expression
Hypotheses cdleme40.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme40.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme40.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme40.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme40.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme40.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme40.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme40.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme40.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme40.i ⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) )
cdleme40.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐷 )
cdleme40a1.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme40a1.c ⊢ 𝐶 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝑌 ) )
cdleme40.t ⊢ 𝑇 = ( ( 𝑣 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdleme40.f ⊢ 𝐹 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑆 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdleme40a1.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) )
cdleme40.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) )
cdleme40.v ⊢ 𝑉 = if ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) , 𝑂 , < )
cdleme40a1.z ⊢ 𝑍 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝐹 ) )
Assertion cdleme40n ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ ⦋ 𝑆 / 𝑢 ⦌ 𝑉 )

Proof

Step Hyp Ref Expression
1 cdleme40.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme40.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme40.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme40.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme40.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme40.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme40.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme40.e ⊢ 𝐸 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9 cdleme40.g ⊢ 𝐺 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10 cdleme40.i ⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐺 ) )
11 cdleme40.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐷 )
12 cdleme40a1.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐸 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
13 cdleme40a1.c ⊢ 𝐶 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝑌 ) )
14 cdleme40.t ⊢ 𝑇 = ( ( 𝑣 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑣 ) ∧ 𝑊 ) ) )
15 cdleme40.f ⊢ 𝐹 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑆 ∨ 𝑣 ) ∧ 𝑊 ) ) )
16 cdleme40a1.x ⊢ 𝑋 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑇 ∨ ( ( 𝑢 ∨ 𝑣 ) ∧ 𝑊 ) ) )
17 cdleme40.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝑋 ) )
18 cdleme40.v ⊢ 𝑉 = if ( 𝑢 ≤ ( 𝑃 ∨ 𝑄 ) , 𝑂 , < )
19 cdleme40a1.z ⊢ 𝑍 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝐹 ) )
20 1 fvexi ⊢ 𝐵 ∈ V
21 nfv ⊢ Ⅎ 𝑣 ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) )
22 nfcv ⊢ Ⅎ 𝑣 ⦋ 𝑅 / 𝑠 ⦌ 𝑁
23 nfra1 ⊢ Ⅎ 𝑣 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝐹 )
24 nfcv ⊢ Ⅎ 𝑣 𝐵
25 23 24 nfriota ⊢ Ⅎ 𝑣 ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝐹 ) )
26 19 25 nfcxfr ⊢ Ⅎ 𝑣 𝑍
27 22 26 nfne ⊢ Ⅎ 𝑣 ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝑍
28 27 a1i ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → Ⅎ 𝑣 ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝑍 )
29 19 a1i ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → 𝑍 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑣 ∈ 𝐴 ( ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑧 = 𝐹 ) ) )
30 neeq2 ⊢ ( 𝐹 = 𝑍 → ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝐹 ↔ ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝑍 ) )
31 30 adantl ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐹 = 𝑍 ) → ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝐹 ↔ ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝑍 ) )
32 simpl11 ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
33 simpl12 ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
34 simpl13 ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
35 simpl21 ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ≠ 𝑄 )
36 simpl22 ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
37 simpl23 ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
38 simpl3 ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≠ 𝑆 ) )
39 simprl ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑣 ∈ 𝐴 )
40 simprrl ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑣 ≤ 𝑊 )
41 simprrr ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) )
42 39 40 41 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) )
43 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 cdleme40m ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ∧ ( 𝑣 ∈ 𝐴 ∧ ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝐹 )
44 32 33 34 35 36 37 38 42 43 syl332anc ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝐹 )
45 44 ex ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ( ( 𝑣 ∈ 𝐴 ∧ ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝐹 ) )
46 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
47 simp23l ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → 𝑆 ∈ 𝐴 )
48 simp23r ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ¬ 𝑆 ≤ 𝑊 )
49 simp21 ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → 𝑃 ≠ 𝑄 )
50 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
51 1 2 3 4 5 6 7 14 15 19 cdleme25cl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑍 ∈ 𝐵 )
52 46 47 48 49 50 51 syl122anc ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → 𝑍 ∈ 𝐵 )
53 simp11 ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
54 simp12 ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
55 simp13 ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
56 2 3 5 6 cdlemb2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑣 ∈ 𝐴 ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) )
57 53 54 55 49 56 syl121anc ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ∃ 𝑣 ∈ 𝐴 ( ¬ 𝑣 ≤ 𝑊 ∧ ¬ 𝑣 ≤ ( 𝑃 ∨ 𝑄 ) ) )
58 21 28 29 31 45 52 57 riotasv3d ⊢ ( ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) ∧ 𝐵 ∈ V ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝑍 )
59 20 58 mpan2 ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ 𝑍 )
60 16 17 18 15 19 cdleme31sn1c ⊢ ( ( 𝑆 ∈ 𝐴 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑆 / 𝑢 ⦌ 𝑉 = 𝑍 )
61 47 50 60 syl2anc ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ⦋ 𝑆 / 𝑢 ⦌ 𝑉 = 𝑍 )
62 59 61 neeqtrrd ⊢ ( ( ( ( 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 ) ∧ 𝑅 ≠ 𝑆 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≠ ⦋ 𝑆 / 𝑢 ⦌ 𝑉 )