Theorem iscvlat 39830

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 6831	. . . 4 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		iscvlat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2794	. . 3 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6831	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5		iscvlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6	4, 5	eqtr4di 2794	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7		fveq2 6831	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8		iscvlat.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
9	7, 8	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
10	9	breqd 5086	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥 ↔ 𝑝 ≤ 𝑥))
11	10	notbid 320	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 ≤ 𝑥))
12		eqidd 2742	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑝 = 𝑝)
13		fveq2 6831	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14		iscvlat.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
15	13, 14	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
16	15	oveqd 7377	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 ∨ 𝑞))
17	12, 9, 16	breq123d 5089	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 ≤ (𝑥 ∨ 𝑞)))
18	11, 17	anbi12d 639	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥 ∧ 𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞))))
19		eqidd 2742	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞)
20	15	oveqd 7377	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 ∨ 𝑝))
21	19, 9, 20	breq123d 5089	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 ≤ (𝑥 ∨ 𝑝)))
22	18, 21	imbi12d 346	. . . . 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 3315	. . . 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 3315	. . 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 3315	. 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 39829	. 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 3634	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 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   class class class wbr 5075  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  lecple 17222  joincjn 18272  Atomscatm 39770  AtLatcal 39771  CvLatclc 39772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-iota 6445  df-fv 6497  df-ov 7363  df-cvlat 39829

This theorem is referenced by:  iscvlat2N  39831  cvlatl  39832  cvlexch1  39835  ishlat2  39860