Theorem dalem18

Description: Lemma for dath . Show that a dummy atom c exists outside of the Y and Z planes (when those planes are equal). This requires that the projective space be 3-dimensional. (Desargues's theorem does not always hold in 2 dimensions.) (Contributed by NM, 29-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 )
	dalem18.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
Assertion	dalem18	⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 )

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		dalem18.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
6	1	dalemkehl	⊢ ( 𝜑 → 𝐾 ∈ HL )
7	1	dalempea	⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
8	1	dalemqea	⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
9	1	dalemrea	⊢ ( 𝜑 → 𝑅 ∈ 𝐴 )
10	3 2 4	3dim3	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
11	6 7 8 9 10	syl13anc	⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
12	5	breq2i	⊢ ( 𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
13	12	notbii	⊢ ( ¬ 𝑐 ≤ 𝑌 ↔ ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
14	13	rexbii	⊢ ( ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 ↔ ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
15	11 14	sylibr	⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 )