Theorem iscvlat 40022

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 6882	. . . 4 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		iscvlat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2822	. . 3 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6882	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5		iscvlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6	4, 5	eqtr4di 2822	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8		iscvlat.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
9	7, 8	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
10	9	breqd 5122	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥 ↔ 𝑝 ≤ 𝑥))
11	10	notbid 321	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 ≤ 𝑥))
12		eqidd 2770	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑝 = 𝑝)
13		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14		iscvlat.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
15	13, 14	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
16	15	oveqd 7428	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 ∨ 𝑞))
17	12, 9, 16	breq123d 5125	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 ≤ (𝑥 ∨ 𝑞)))
18	11, 17	anbi12d 643	. . . . . 6 ⊢ (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥 ∧ 𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 ≤ 𝑥 ∧ 𝑝 ≤ (𝑥 ∨ 𝑞))))
19		eqidd 2770	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑞 = 𝑞)
20	15	oveqd 7428	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 ∨ 𝑝))
21	19, 9, 20	breq123d 5125	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 ≤ (𝑥 ∨ 𝑝)))
22	18, 21	imbi12d 347	. . . . 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 3344	. . . 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 3344	. . 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 3344	. 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 40021	. 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 3661	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 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5111  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  lecple 17317  joincjn 18367  Atomscatm 39962  AtLatcal 39963  CvLatclc 39964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4491  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5112  df-iota 6493  df-fv 6545  df-ov 7414  df-cvlat 40021

This theorem is referenced by:  iscvlat2N  40023  cvlatl  40024  cvlexch1  40027  ishlat2  40052