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Theorem iscvlat 36451
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 6663 . . . 4 (𝑘 = 𝐾 → (Atoms‘𝑘) = (Atoms‘𝐾))
2 iscvlat.a . . . 4 𝐴 = (Atoms‘𝐾)
31, 2syl6eqr 2872 . . 3 (𝑘 = 𝐾 → (Atoms‘𝑘) = 𝐴)
4 fveq2 6663 . . . . . 6 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
5 iscvlat.b . . . . . 6 𝐵 = (Base‘𝐾)
64, 5syl6eqr 2872 . . . . 5 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
7 fveq2 6663 . . . . . . . . . 10 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
8 iscvlat.l . . . . . . . . . 10 = (le‘𝐾)
97, 8syl6eqr 2872 . . . . . . . . 9 (𝑘 = 𝐾 → (le‘𝑘) = )
109breqd 5068 . . . . . . . 8 (𝑘 = 𝐾 → (𝑝(le‘𝑘)𝑥𝑝 𝑥))
1110notbid 320 . . . . . . 7 (𝑘 = 𝐾 → (¬ 𝑝(le‘𝑘)𝑥 ↔ ¬ 𝑝 𝑥))
12 eqidd 2820 . . . . . . . 8 (𝑘 = 𝐾𝑝 = 𝑝)
13 fveq2 6663 . . . . . . . . . 10 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
14 iscvlat.j . . . . . . . . . 10 = (join‘𝐾)
1513, 14syl6eqr 2872 . . . . . . . . 9 (𝑘 = 𝐾 → (join‘𝑘) = )
1615oveqd 7165 . . . . . . . 8 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑞) = (𝑥 𝑞))
1712, 9, 16breq123d 5071 . . . . . . 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 2820 . . . . . . 7 (𝑘 = 𝐾𝑞 = 𝑞)
2015oveqd 7165 . . . . . . 7 (𝑘 = 𝐾 → (𝑥(join‘𝑘)𝑝) = (𝑥 𝑝))
2119, 9, 20breq123d 5071 . . . . . 6 (𝑘 = 𝐾 → (𝑞(le‘𝑘)(𝑥(join‘𝑘)𝑝) ↔ 𝑞 (𝑥 𝑝)))
2218, 21imbi12d 347 . . . . 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 3400 . . . 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 3400 . . 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 3400 . 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 36450 . 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 3681 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 208  wa 398   = wceq 1530  wcel 2107  wral 3136   class class class wbr 5057  cfv 6348  (class class class)co 7148  Basecbs 16475  lecple 16564  joincjn 17546  Atomscatm 36391  AtLatcal 36392  CvLatclc 36393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7151  df-cvlat 36450
This theorem is referenced by:  iscvlat2N  36452  cvlatl  36453  cvlexch1  36456  ishlat2  36481
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