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 )