Theorem iscvlat 39324

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

Hypotheses

Ref	Expression
iscvlat.b	⊢ 𝐵 = (Base‘𝐾)
iscvlat.l	⊢ ≤ = (le‘𝐾)
iscvlat.j	⊢ ∨ = (join‘𝐾)
iscvlat.a	⊢ 𝐴 = (Atoms‘𝐾)

Assertion

Ref	Expression
iscvlat	⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝))))

Distinct variable groups:   𝑞,𝑝,𝐴   𝑥,𝐵   𝑥,𝑝,𝐾,𝑞

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑞,𝑝)   ∨ (𝑥,𝑞,𝑝)   ≤ (𝑥,𝑞,𝑝)

Proof of Theorem iscvlat

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6906	. . . 4 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		iscvlat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2795	. . 3 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6906	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5		iscvlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6	4, 5	eqtr4di 2795	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8		iscvlat.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
9	7, 8	eqtr4di 2795	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
10	9	breqd 5154	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥 ↔ 𝑝 ≤ 𝑥))
11	10	notbid 318	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 ≤ 𝑥))
12		eqidd 2738	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑝 = 𝑝)
13		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14		iscvlat.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
15	13, 14	eqtr4di 2795	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
16	15	oveqd 7448	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 ∨ 𝑞))
17	12, 9, 16	breq123d 5157	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 ≤ (𝑥 ∨ 𝑞)))
18	11, 17	anbi12d 632	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥 ∧ 𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞))))
19		eqidd 2738	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞)
20	15	oveqd 7448	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 ∨ 𝑝))
21	19, 9, 20	breq123d 5157	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 ≤ (𝑥 ∨ 𝑝)))
22	18, 21	imbi12d 344	. . . . 5 ⊢ (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	6, 22	raleqbidv 3346	. . . 4 ⊢ (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	3, 23	raleqbidv 3346	. . 3 ⊢ (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	3, 24	raleqbidv 3346	. 2 ⊢ (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 39323	. 2 ⊢ 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 3695	1 ⊢ (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝 ∈ 𝐴 ∀𝑞 ∈ 𝐴 ∀𝑥 ∈ 𝐵 ((¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞)) → 𝑞 ≤ (𝑥 ∨ 𝑝))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  joincjn 18357  Atomscatm 39264  AtLatcal 39265  CvLatclc 39266

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-cvlat 39323

This theorem is referenced by:  iscvlat2N  39325  cvlatl  39326  cvlexch1  39329  ishlat2  39354