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Theorem iscvlat 38827
Description: The predicate "is an atomic lattice with the covering (or exchange) property". (Contributed by NM, 5-Nov-2012.)
Hypotheses
Ref Expression
iscvlat.b 𝐡 = (Baseβ€˜πΎ)
iscvlat.l ≀ = (leβ€˜πΎ)
iscvlat.j ∨ = (joinβ€˜πΎ)
iscvlat.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
iscvlat (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
Distinct variable groups:   π‘ž,𝑝,𝐴   π‘₯,𝐡   π‘₯,𝑝,𝐾,π‘ž
Allowed substitution hints:   𝐴(π‘₯)   𝐡(π‘ž,𝑝)   ∨ (π‘₯,π‘ž,𝑝)   ≀ (π‘₯,π‘ž,𝑝)

Proof of Theorem iscvlat
Dummy variable π‘˜ is distinct from all other variables.
StepHypRef Expression
1 fveq2 6902 . . . 4 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = (Atomsβ€˜πΎ))
2 iscvlat.a . . . 4 𝐴 = (Atomsβ€˜πΎ)
31, 2eqtr4di 2786 . . 3 (π‘˜ = 𝐾 β†’ (Atomsβ€˜π‘˜) = 𝐴)
4 fveq2 6902 . . . . . 6 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = (Baseβ€˜πΎ))
5 iscvlat.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
64, 5eqtr4di 2786 . . . . 5 (π‘˜ = 𝐾 β†’ (Baseβ€˜π‘˜) = 𝐡)
7 fveq2 6902 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = (leβ€˜πΎ))
8 iscvlat.l . . . . . . . . . 10 ≀ = (leβ€˜πΎ)
97, 8eqtr4di 2786 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (leβ€˜π‘˜) = ≀ )
109breqd 5163 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (𝑝(leβ€˜π‘˜)π‘₯ ↔ 𝑝 ≀ π‘₯))
1110notbid 317 . . . . . . 7 (π‘˜ = 𝐾 β†’ (Β¬ 𝑝(leβ€˜π‘˜)π‘₯ ↔ Β¬ 𝑝 ≀ π‘₯))
12 eqidd 2729 . . . . . . . 8 (π‘˜ = 𝐾 β†’ 𝑝 = 𝑝)
13 fveq2 6902 . . . . . . . . . 10 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = (joinβ€˜πΎ))
14 iscvlat.j . . . . . . . . . 10 ∨ = (joinβ€˜πΎ)
1513, 14eqtr4di 2786 . . . . . . . . 9 (π‘˜ = 𝐾 β†’ (joinβ€˜π‘˜) = ∨ )
1615oveqd 7443 . . . . . . . 8 (π‘˜ = 𝐾 β†’ (π‘₯(joinβ€˜π‘˜)π‘ž) = (π‘₯ ∨ π‘ž))
1712, 9, 16breq123d 5166 . . . . . . 7 (π‘˜ = 𝐾 β†’ (𝑝(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)π‘ž) ↔ 𝑝 ≀ (π‘₯ ∨ π‘ž)))
1811, 17anbi12d 630 . . . . . 6 (π‘˜ = 𝐾 β†’ ((Β¬ 𝑝(leβ€˜π‘˜)π‘₯ ∧ 𝑝(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)π‘ž)) ↔ (Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž))))
19 eqidd 2729 . . . . . . 7 (π‘˜ = 𝐾 β†’ π‘ž = π‘ž)
2015oveqd 7443 . . . . . . 7 (π‘˜ = 𝐾 β†’ (π‘₯(joinβ€˜π‘˜)𝑝) = (π‘₯ ∨ 𝑝))
2119, 9, 20breq123d 5166 . . . . . 6 (π‘˜ = 𝐾 β†’ (π‘ž(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑝) ↔ π‘ž ≀ (π‘₯ ∨ 𝑝)))
2218, 21imbi12d 343 . . . . 5 (π‘˜ = 𝐾 β†’ (((Β¬ 𝑝(leβ€˜π‘˜)π‘₯ ∧ 𝑝(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)π‘ž)) β†’ π‘ž(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑝)) ↔ ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
236, 22raleqbidv 3340 . . . 4 (π‘˜ = 𝐾 β†’ (βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)((Β¬ 𝑝(leβ€˜π‘˜)π‘₯ ∧ 𝑝(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)π‘ž)) β†’ π‘ž(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑝)) ↔ βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
243, 23raleqbidv 3340 . . 3 (π‘˜ = 𝐾 β†’ (βˆ€π‘ž ∈ (Atomsβ€˜π‘˜)βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)((Β¬ 𝑝(leβ€˜π‘˜)π‘₯ ∧ 𝑝(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)π‘ž)) β†’ π‘ž(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑝)) ↔ βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
253, 24raleqbidv 3340 . 2 (π‘˜ = 𝐾 β†’ (βˆ€π‘ ∈ (Atomsβ€˜π‘˜)βˆ€π‘ž ∈ (Atomsβ€˜π‘˜)βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)((Β¬ 𝑝(leβ€˜π‘˜)π‘₯ ∧ 𝑝(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)π‘ž)) β†’ π‘ž(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑝)) ↔ βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
26 df-cvlat 38826 . 2 CvLat = {π‘˜ ∈ AtLat ∣ βˆ€π‘ ∈ (Atomsβ€˜π‘˜)βˆ€π‘ž ∈ (Atomsβ€˜π‘˜)βˆ€π‘₯ ∈ (Baseβ€˜π‘˜)((Β¬ 𝑝(leβ€˜π‘˜)π‘₯ ∧ 𝑝(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)π‘ž)) β†’ π‘ž(leβ€˜π‘˜)(π‘₯(joinβ€˜π‘˜)𝑝))}
2725, 26elrab2 3687 1 (𝐾 ∈ CvLat ↔ (𝐾 ∈ AtLat ∧ βˆ€π‘ ∈ 𝐴 βˆ€π‘ž ∈ 𝐴 βˆ€π‘₯ ∈ 𝐡 ((Β¬ 𝑝 ≀ π‘₯ ∧ 𝑝 ≀ (π‘₯ ∨ π‘ž)) β†’ π‘ž ≀ (π‘₯ ∨ 𝑝))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058   class class class wbr 5152  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187  lecple 17247  joincjn 18310  Atomscatm 38767  AtLatcal 38768  CvLatclc 38769
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429  df-cvlat 38826
This theorem is referenced by:  iscvlat2N  38828  cvlatl  38829  cvlexch1  38832  ishlat2  38857
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