Theorem iscvlat 39279

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 6920	. . . 4 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		iscvlat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2798	. . 3 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6920	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5		iscvlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6	4, 5	eqtr4di 2798	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8		iscvlat.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
9	7, 8	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
10	9	breqd 5177	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥 ↔ 𝑝 ≤ 𝑥))
11	10	notbid 318	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 ≤ 𝑥))
12		eqidd 2741	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑝 = 𝑝)
13		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14		iscvlat.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
15	13, 14	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
16	15	oveqd 7465	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 ∨ 𝑞))
17	12, 9, 16	breq123d 5180	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 ≤ (𝑥 ∨ 𝑞)))
18	11, 17	anbi12d 631	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥 ∧ 𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞))))
19		eqidd 2741	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞)
20	15	oveqd 7465	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 ∨ 𝑝))
21	19, 9, 20	breq123d 5180	. . . . . 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 3354	. . . 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 3354	. . 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 3354	. 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 39278	. 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 3711	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 1537   ∈ wcel 2108  ∀wral 3067   class class class wbr 5166  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  lecple 17318  joincjn 18381  Atomscatm 39219  AtLatcal 39220  CvLatclc 39221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451  df-cvlat 39278

This theorem is referenced by:  iscvlat2N  39280  cvlatl  39281  cvlexch1  39284  ishlat2  39309