Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  iscvlat Structured version   Visualization version   GIF version

Theorem iscvlat 35133
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 6333 . . . 4 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 iscvlat.a . . . 4 𝐴 = (Atoms‘𝐾)
31, 2syl6eqr 2823 . . 3 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6333 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5 iscvlat.b . . . . . 6 𝐵 = (Base‘𝐾)
64, 5syl6eqr 2823 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7 fveq2 6333 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8 iscvlat.l . . . . . . . . . 10 = (le‘𝐾)
97, 8syl6eqr 2823 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = )
109breqd 4798 . . . . . . . 8 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥𝑝 𝑥))
1110notbid 307 . . . . . . 7 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 𝑥))
12 eqidd 2772 . . . . . . . 8 (𝑘 = 𝐾𝑝 = 𝑝)
13 fveq2 6333 . . . . . . . . . 10 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 iscvlat.j . . . . . . . . . 10 = (join‘𝐾)
1513, 14syl6eqr 2823 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = )
1615oveqd 6811 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 𝑞))
1712, 9, 16breq123d 4801 . . . . . . 7 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 (𝑥 𝑞)))
1811, 17anbi12d 610 . . . . . 6 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 𝑥𝑝 (𝑥 𝑞))))
19 eqidd 2772 . . . . . . 7 (𝑘 = 𝐾𝑞 = 𝑞)
2015oveqd 6811 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 𝑝))
2119, 9, 20breq123d 4801 . . . . . 6 (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 (𝑥 𝑝)))
2218, 21imbi12d 333 . . . . 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 3301 . . . 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 3301 . . 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 3301 . 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 35132 . 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 3519 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 196  wa 382   = wceq 1631  wcel 2145  wral 3061   class class class wbr 4787  cfv 6032  (class class class)co 6794  Basecbs 16065  lecple 16157  joincjn 17153  Atomscatm 35073  AtLatcal 35074  CvLatclc 35075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5995  df-fv 6040  df-ov 6797  df-cvlat 35132
This theorem is referenced by:  iscvlat2N  35134  cvlatl  35135  cvlexch1  35138  ishlat2  35163
  Copyright terms: Public domain W3C validator