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Theorem islat 18491
Description: The predicate "is a lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
islat.b 𝐵 = (Base‘𝐾)
islat.j = (join‘𝐾)
islat.m = (meet‘𝐾)
Assertion
Ref Expression
islat (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))

Proof of Theorem islat
Dummy variable 𝑙 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . . 6 (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2 islat.j . . . . . 6 = (join‘𝐾)
31, 2eqtr4di 2793 . . . . 5 (𝑙 = 𝐾 → (join‘𝑙) = )
43dmeqd 5919 . . . 4 (𝑙 = 𝐾 → dom (join‘𝑙) = dom )
5 fveq2 6907 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 islat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2793 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87sqxpeqd 5721 . . . 4 (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵))
94, 8eqeq12d 2751 . . 3 (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom = (𝐵 × 𝐵)))
10 fveq2 6907 . . . . . 6 (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11 islat.m . . . . . 6 = (meet‘𝐾)
1210, 11eqtr4di 2793 . . . . 5 (𝑙 = 𝐾 → (meet‘𝑙) = )
1312dmeqd 5919 . . . 4 (𝑙 = 𝐾 → dom (meet‘𝑙) = dom )
1413, 8eqeq12d 2751 . . 3 (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom = (𝐵 × 𝐵)))
159, 14anbi12d 632 . 2 (𝑙 = 𝐾 → ((dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙))) ↔ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
16 df-lat 18490 . 2 Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
1715, 16elrab2 3698 1 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106   × cxp 5687  dom cdm 5689  cfv 6563  Basecbs 17245  Posetcpo 18365  joincjn 18369  meetcmee 18370  Latclat 18489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-lat 18490
This theorem is referenced by:  odulatb  18492  latcl2  18494  latlem  18495  latpos  18496  latjcom  18505  latmcom  18521  clatl  18566  toslat  48771
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