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Theorem iscvlat 35344
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 ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (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.
StepHypRef Expression
1 fveq2 6411 . . . 4 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 iscvlat.a . . . 4 𝐴 = (Atoms‘𝐾)
31, 2syl6eqr 2851 . . 3 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6411 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5 iscvlat.b . . . . . 6 𝐵 = (Base‘𝐾)
64, 5syl6eqr 2851 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7 fveq2 6411 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8 iscvlat.l . . . . . . . . . 10 = (le‘𝐾)
97, 8syl6eqr 2851 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = )
109breqd 4854 . . . . . . . 8 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥𝑝 𝑥))
1110notbid 310 . . . . . . 7 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 𝑥))
12 eqidd 2800 . . . . . . . 8 (𝑘 = 𝐾𝑝 = 𝑝)
13 fveq2 6411 . . . . . . . . . 10 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 iscvlat.j . . . . . . . . . 10 = (join‘𝐾)
1513, 14syl6eqr 2851 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = )
1615oveqd 6895 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 𝑞))
1712, 9, 16breq123d 4857 . . . . . . 7 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 (𝑥 𝑞)))
1811, 17anbi12d 625 . . . . . 6 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 𝑥𝑝 (𝑥 𝑞))))
19 eqidd 2800 . . . . . . 7 (𝑘 = 𝐾𝑞 = 𝑞)
2015oveqd 6895 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 𝑝))
2119, 9, 20breq123d 4857 . . . . . 6 (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 (𝑥 𝑝)))
2218, 21imbi12d 336 . . . . 5 (𝑘 = 𝐾 → (((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
236, 22raleqbidv 3335 . . . 4 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ∀𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
243, 23raleqbidv 3335 . . 3 (𝑘 = 𝐾 → (∀𝑞 ∈ (Atoms‘𝑘)∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ∀𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
253, 24raleqbidv 3335 . 2 (𝑘 = 𝐾 → (∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝)) ↔ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
26 df-cvlat 35343 . 2 CvLat = {𝑘 ∈ AtLat ∣ ∀𝑝 ∈ (Atoms‘𝑘)∀𝑞 ∈ (Atoms‘𝑘)∀𝑥 ∈ (Base‘𝑘)((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) → 𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝))}
2725, 26elrab2 3560 1 (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ ∀𝑝𝐴𝑞𝐴𝑥𝐵 ((¬ 𝑝 𝑥𝑝 (𝑥 𝑞)) → 𝑞 (𝑥 𝑝))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089   class class class wbr 4843  cfv 6101  (class class class)co 6878  Basecbs 16184  lecple 16274  joincjn 17259  Atomscatm 35284  AtLatcal 35285  CvLatclc 35286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109  df-ov 6881  df-cvlat 35343
This theorem is referenced by:  iscvlat2N  35345  cvlatl  35346  cvlexch1  35349  ishlat2  35374
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