Theorem dalem15

Description: Lemma for dath . The axis of perspectivity X is a line. (Contributed by NM, 21-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 )
	dalem15.m	⊢ ∧ = ( meet ‘ 𝐾 )
	dalem15.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
	dalem15.o	⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
	dalem15.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
	dalem15.z	⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
	dalem15.x	⊢ 𝑋 = ( 𝑌 ∧ 𝑍 )
Assertion	dalem15	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑋 ∈ 𝑁 )

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		dalem15.m	⊢ ∧ = ( meet ‘ 𝐾 )
6		dalem15.n	⊢ 𝑁 = ( LLines ‘ 𝐾 )
7		dalem15.o	⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
8		dalem15.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
9		dalem15.z	⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
10		dalem15.x	⊢ 𝑋 = ( 𝑌 ∧ 𝑍 )
11		eqid	⊢ ( LVols ‘ 𝐾 ) = ( LVols ‘ 𝐾 )
12		eqid	⊢ ( 𝑌 ∨ 𝐶 ) = ( 𝑌 ∨ 𝐶 )
13	1 2 3 4 7 11 8 9 12	dalem14	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∨ 𝑍 ) ∈ ( LVols ‘ 𝐾 ) )
14	1	dalemkehl	⊢ ( 𝜑 → 𝐾 ∈ HL )
15	1	dalemyeo	⊢ ( 𝜑 → 𝑌 ∈ 𝑂 )
16	1	dalemzeo	⊢ ( 𝜑 → 𝑍 ∈ 𝑂 )
17	3 5 6 7 11	2lplnmj	⊢ ( ( 𝐾 ∈ HL ∧ 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) → ( ( 𝑌 ∧ 𝑍 ) ∈ 𝑁 ↔ ( 𝑌 ∨ 𝑍 ) ∈ ( LVols ‘ 𝐾 ) ) )
18	14 15 16 17	syl3anc	⊢ ( 𝜑 → ( ( 𝑌 ∧ 𝑍 ) ∈ 𝑁 ↔ ( 𝑌 ∨ 𝑍 ) ∈ ( LVols ‘ 𝐾 ) ) )
19	18	adantr	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( ( 𝑌 ∧ 𝑍 ) ∈ 𝑁 ↔ ( 𝑌 ∨ 𝑍 ) ∈ ( LVols ‘ 𝐾 ) ) )
20	13 19	mpbird	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∧ 𝑍 ) ∈ 𝑁 )
21	10 20	eqeltrid	⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝑋 ∈ 𝑁 )