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Theorem iscvlat 39304
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 6906 . . . 4 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 iscvlat.a . . . 4 𝐴 = (Atoms‘𝐾)
31, 2eqtr4di 2792 . . 3 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6906 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5 iscvlat.b . . . . . 6 𝐵 = (Base‘𝐾)
64, 5eqtr4di 2792 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7 fveq2 6906 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8 iscvlat.l . . . . . . . . . 10 = (le‘𝐾)
97, 8eqtr4di 2792 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = )
109breqd 5158 . . . . . . . 8 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥𝑝 𝑥))
1110notbid 318 . . . . . . 7 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 𝑥))
12 eqidd 2735 . . . . . . . 8 (𝑘 = 𝐾𝑝 = 𝑝)
13 fveq2 6906 . . . . . . . . . 10 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 iscvlat.j . . . . . . . . . 10 = (join‘𝐾)
1513, 14eqtr4di 2792 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = )
1615oveqd 7447 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 𝑞))
1712, 9, 16breq123d 5161 . . . . . . 7 (𝑘 = 𝐾 → (𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞) ↔ 𝑝 (𝑥 𝑞)))
1811, 17anbi12d 632 . . . . . 6 (𝑘 = 𝐾 → ((¬ 𝑝(le‘𝑘)𝑥𝑝(le‘𝑘)(𝑥(join‘𝑘)𝑞)) ↔ (¬ 𝑝 𝑥𝑝 (𝑥 𝑞))))
19 eqidd 2735 . . . . . . 7 (𝑘 = 𝐾𝑞 = 𝑞)
2015oveqd 7447 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 𝑝))
2119, 9, 20breq123d 5161 . . . . . 6 (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 (𝑥 𝑝)))
2218, 21imbi12d 344 . . . . 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 3343 . . . 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 3343 . . 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 3343 . 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 39303 . 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 3697 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 206  wa 395   = wceq 1536  wcel 2105  wral 3058   class class class wbr 5147  cfv 6562  (class class class)co 7430  Basecbs 17244  lecple 17304  joincjn 18368  Atomscatm 39244  AtLatcal 39245  CvLatclc 39246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-iota 6515  df-fv 6570  df-ov 7433  df-cvlat 39303
This theorem is referenced by:  iscvlat2N  39305  cvlatl  39306  cvlexch1  39309  ishlat2  39334
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