Theorem dalemsly

Description: Lemma for dath . Frequently-used utility lemma. (Contributed by NM, 15-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 )
	dalemsly.z	⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
Assertion	dalemsly	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑆 ≤ 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		dalemsly.z	⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 )
6	1	dalemkelat	⊢ ( 𝜑 → 𝐾 ∈ Lat )
7	1 4	dalemseb	⊢ ( 𝜑 → 𝑆 ∈ ( Base ‘ 𝐾 ) )
8	1 3 4	dalemtjueb	⊢ ( 𝜑 → ( 𝑇 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
9		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
10	9 2 3	latlej1	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑇 ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑆 ≤ ( 𝑆 ∨ ( 𝑇 ∨ 𝑈 ) ) )
11	6 7 8 10	syl3anc	⊢ ( 𝜑 → 𝑆 ≤ ( 𝑆 ∨ ( 𝑇 ∨ 𝑈 ) ) )
12	1	dalemkehl	⊢ ( 𝜑 → 𝐾 ∈ HL )
13	1	dalemsea	⊢ ( 𝜑 → 𝑆 ∈ 𝐴 )
14	1	dalemtea	⊢ ( 𝜑 → 𝑇 ∈ 𝐴 )
15	1	dalemuea	⊢ ( 𝜑 → 𝑈 ∈ 𝐴 )
16	3 4	hlatjass	⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) = ( 𝑆 ∨ ( 𝑇 ∨ 𝑈 ) ) )
17	12 13 14 15 16	syl13anc	⊢ ( 𝜑 → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) = ( 𝑆 ∨ ( 𝑇 ∨ 𝑈 ) ) )
18	11 17	breqtrrd	⊢ ( 𝜑 → 𝑆 ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
19	18 5	breqtrrdi	⊢ ( 𝜑 → 𝑆 ≤ 𝑍 )
20	19	adantr	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑆 ≤ 𝑍 )
21		simpr	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑌 = 𝑍 )
22	20 21	breqtrrd	⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑆 ≤ 𝑌 )