Theorem iscvlat

Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012)

	Ref	Expression
Hypotheses	iscvlat.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	iscvlat.l	⊢ ≤ = ( le ‘ 𝐾 )
	iscvlat.j	⊢ ∨ = ( join ‘ 𝐾 )
	iscvlat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	iscvlat	⊢ ( 𝐾 ∈ CvLat ↔ ( 𝐾 ∈ AtLat ∧ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ∀ 𝑥 ∈ 𝐵 ( ( ¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) → 𝑞 ≤ ( 𝑥 ∨ 𝑝 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		iscvlat.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		iscvlat.l	⊢ ≤ = ( le ‘ 𝐾 )
3		iscvlat.j	⊢ ∨ = ( join ‘ 𝐾 )
4		iscvlat.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5		fveq2	⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = ( Atoms ‘ 𝐾 ) )
6	5 4	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( Atoms ‘ 𝑘 ) = 𝐴 )
7		fveq2	⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = ( Base ‘ 𝐾 ) )
8	7 1	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( Base ‘ 𝑘 ) = 𝐵 )
9		fveq2	⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ( le ‘ 𝐾 ) )
10	9 2	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( le ‘ 𝑘 ) = ≤ )
11	10	breqd	⊢ ( 𝑘 = 𝐾 → ( 𝑝 ( le ‘ 𝑘 ) 𝑥 ↔ 𝑝 ≤ 𝑥 ) )
12	11	notbid	⊢ ( 𝑘 = 𝐾 → ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑥 ↔ ¬ 𝑝 ≤ 𝑥 ) )
13		eqidd	⊢ ( 𝑘 = 𝐾 → 𝑝 = 𝑝 )
14		fveq2	⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ( join ‘ 𝐾 ) )
15	14 3	eqtr4di	⊢ ( 𝑘 = 𝐾 → ( join ‘ 𝑘 ) = ∨ )
16	15	oveqd	⊢ ( 𝑘 = 𝐾 → ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) = ( 𝑥 ∨ 𝑞 ) )
17	13 10 16	breq123d	⊢ ( 𝑘 = 𝐾 → ( 𝑝 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) ↔ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) )
18	12 17	anbi12d	⊢ ( 𝑘 = 𝐾 → ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑥 ∧ 𝑝 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) ) ↔ ( ¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) ) )
19		eqidd	⊢ ( 𝑘 = 𝐾 → 𝑞 = 𝑞 )
20	15	oveqd	⊢ ( 𝑘 = 𝐾 → ( 𝑥 ( join ‘ 𝑘 ) 𝑝 ) = ( 𝑥 ∨ 𝑝 ) )
21	19 10 20	breq123d	⊢ ( 𝑘 = 𝐾 → ( 𝑞 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑝 ) ↔ 𝑞 ≤ ( 𝑥 ∨ 𝑝 ) ) )
22	18 21	imbi12d	⊢ ( 𝑘 = 𝐾 → ( ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑥 ∧ 𝑝 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) ) → 𝑞 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑝 ) ) ↔ ( ( ¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) → 𝑞 ≤ ( 𝑥 ∨ 𝑝 ) ) ) )
23	8 22	raleqbidv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑥 ∈ ( Base ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑥 ∧ 𝑝 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) ) → 𝑞 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑝 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ( ( ¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) → 𝑞 ≤ ( 𝑥 ∨ 𝑝 ) ) ) )
24	6 23	raleqbidv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑥 ∈ ( Base ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑥 ∧ 𝑝 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) ) → 𝑞 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑝 ) ) ↔ ∀ 𝑞 ∈ 𝐴 ∀ 𝑥 ∈ 𝐵 ( ( ¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) → 𝑞 ≤ ( 𝑥 ∨ 𝑝 ) ) ) )
25	6 24	raleqbidv	⊢ ( 𝑘 = 𝐾 → ( ∀ 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑥 ∈ ( Base ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑥 ∧ 𝑝 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) ) → 𝑞 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑝 ) ) ↔ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ∀ 𝑥 ∈ 𝐵 ( ( ¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) → 𝑞 ≤ ( 𝑥 ∨ 𝑝 ) ) ) )
26		df-cvlat	⊢ CvLat = { 𝑘 ∈ AtLat ∣ ∀ 𝑝 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑞 ∈ ( Atoms ‘ 𝑘 ) ∀ 𝑥 ∈ ( Base ‘ 𝑘 ) ( ( ¬ 𝑝 ( le ‘ 𝑘 ) 𝑥 ∧ 𝑝 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑞 ) ) → 𝑞 ( le ‘ 𝑘 ) ( 𝑥 ( join ‘ 𝑘 ) 𝑝 ) ) }
27	25 26	elrab2	⊢ ( 𝐾 ∈ CvLat ↔ ( 𝐾 ∈ AtLat ∧ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ∀ 𝑥 ∈ 𝐵 ( ( ¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ ( 𝑥 ∨ 𝑞 ) ) → 𝑞 ≤ ( 𝑥 ∨ 𝑝 ) ) ) )