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Metamath Proof Explorer

Theorem dalem9

Description: Lemma for dath . Since -. C .<_ Y , the join Y .\/ C forms a 3-dimensional space. (Contributed by NM, 20-Jul-2012)

Ref Expression
Hypotheses dalema.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) )
dalemc.l ⊢ ≤ = ( le ‘ 𝐾 )
dalemc.j ⊢ ∨ = ( join ‘ 𝐾 )
dalemc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dalem9.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
dalem9.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
dalem9.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
dalem9.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
dalem9.w ⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
Assertion dalem9 ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑊 ∈ 𝑉 )

Proof

Step Hyp Ref Expression
1 dalema.ph ⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) )
2 dalemc.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dalemc.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dalemc.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 dalem9.o ⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6 dalem9.v ⊢ 𝑉 = ( LVols ‘ 𝐾 )
7 dalem9.y ⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
8 dalem9.z ⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
9 dalem9.w ⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
10 1 dalemkehl ⊢ ( 𝜑 → 𝐾 ∈ HL )
11 10 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐾 ∈ HL )
12 1 dalemyeo ⊢ ( 𝜑 → 𝑌 ∈ 𝑂 )
13 12 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑌 ∈ 𝑂 )
14 1 2 3 4 5 7 dalemcea ⊢ ( 𝜑 → 𝐶 ∈ 𝐴 )
15 14 adantr ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐶 ∈ 𝐴 )
16 1 2 3 4 5 7 8 dalem-cly ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ¬ 𝐶 ≤ 𝑌 )
17 2 3 4 5 6 lvoli3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝐶 ∈ 𝐴 ) ∧ ¬ 𝐶 ≤ 𝑌 ) → ( 𝑌 ∨ 𝐶 ) ∈ 𝑉 )
18 11 13 15 16 17 syl31anc ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝐶 ) ∈ 𝑉 )
19 9 18 eqeltrid ⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑊 ∈ 𝑉 )