dalemply

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 13-Aug-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 )
	dalempnes.o	⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
	dalempnes.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
Assertion	dalemply	⊢ ( 𝜑 → 𝑃 ≤ 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		dalempnes.o	⊢ 𝑂 = ( LPlanes ‘ 𝐾 )
6		dalempnes.y	⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 )
7	1	dalemkelat	⊢ ( 𝜑 → 𝐾 ∈ Lat )
8	1 4	dalempeb	⊢ ( 𝜑 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
9	1	dalemkehl	⊢ ( 𝜑 → 𝐾 ∈ HL )
10	1	dalemqea	⊢ ( 𝜑 → 𝑄 ∈ 𝐴 )
11	1	dalemrea	⊢ ( 𝜑 → 𝑅 ∈ 𝐴 )
12		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
13	12 3 4	hlatjcl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
14	9 10 11 13	syl3anc	⊢ ( 𝜑 → ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
15	12 2 3	latlej1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) )
16	7 8 14 15	syl3anc	⊢ ( 𝜑 → 𝑃 ≤ ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) )
17	1	dalempea	⊢ ( 𝜑 → 𝑃 ∈ 𝐴 )
18	3 4	hlatjass	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) )
19	9 17 10 11 18	syl13anc	⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) = ( 𝑃 ∨ ( 𝑄 ∨ 𝑅 ) ) )
20	16 19	breqtrrd	⊢ ( 𝜑 → 𝑃 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) )
21	20 6	breqtrrdi	⊢ ( 𝜑 → 𝑃 ≤ 𝑌 )

Metamath Proof Explorer

Theorem dalemply

Proof