Theorem iscvlat 39728

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 6844	. . . 4 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2		iscvlat.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
3	1, 2	eqtr4di 2790	. . 3 ⊢ (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4		fveq2 6844	. . . . . 6 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5		iscvlat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
6	4, 5	eqtr4di 2790	. . . . 5 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8		iscvlat.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
9	7, 8	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
10	9	breqd 5111	. . . . . . . 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 6844	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14		iscvlat.j	. . . . . . . . . 10 ⊢ ∨ = (join‘𝐾)
15	13, 14	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
16	15	oveqd 7387	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 ∨ 𝑞))
17	12, 9, 16	breq123d 5114	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 ≤ (𝑥 ∨ 𝑞)))
18	11, 17	anbi12d 633	. . . . . 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 7387	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 ∨ 𝑝))
21	19, 9, 20	breq123d 5114	. . . . . 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 3318	. . . 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 3318	. . 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 3318	. 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 39727	. 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 3651	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 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5100  ‘cfv 6502  (class class class)co 7370  Basecbs 17150  lecple 17198  joincjn 18248  Atomscatm 39668  AtLatcal 39669  CvLatclc 39670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6458  df-fv 6510  df-ov 7373  df-cvlat 39727

This theorem is referenced by:  iscvlat2N  39729  cvlatl  39730  cvlexch1  39733  ishlat2  39758