Theorem dalem14

Description: Lemma for dath . Planes Y and Z form a 3-dimensional space (when they are different). (Contributed by NM, 22-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 )
	dalem14.o	⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
	dalem14.v	⊢ 𝑉 = ( LVols ‘ 𝐾 )
	dalem14.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
	dalem14.z	⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
	dalem14.w	⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
Assertion	dalem14	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝑍 ) ∈ 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		dalem14.o	⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6		dalem14.v	⊢ 𝑉 = ( LVols ‘ 𝐾 )
7		dalem14.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
8		dalem14.z	⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
9		dalem14.w	⊢ 𝑊 = ( 𝑌 ∨ 𝐶 )
10	1 2 3 4 5 7 8 9	dalem13	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝑍 ) = 𝑊 )
11	1 2 3 4 5 6 7 8 9	dalem9	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑊 ∈ 𝑉 )
12	10 11	eqeltrd	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝑍 ) ∈ 𝑉 )