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Theorem islat 18478
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 6906 . . . . . 6 (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2 islat.j . . . . . 6 = (join‘𝐾)
31, 2eqtr4di 2795 . . . . 5 (𝑙 = 𝐾 → (join‘𝑙) = )
43dmeqd 5916 . . . 4 (𝑙 = 𝐾 → dom (join‘𝑙) = dom )
5 fveq2 6906 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 islat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2795 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87sqxpeqd 5717 . . . 4 (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵))
94, 8eqeq12d 2753 . . 3 (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom = (𝐵 × 𝐵)))
10 fveq2 6906 . . . . . 6 (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11 islat.m . . . . . 6 = (meet‘𝐾)
1210, 11eqtr4di 2795 . . . . 5 (𝑙 = 𝐾 → (meet‘𝑙) = )
1312dmeqd 5916 . . . 4 (𝑙 = 𝐾 → dom (meet‘𝑙) = dom )
1413, 8eqeq12d 2753 . . 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 18477 . 2 Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
1715, 16elrab2 3695 1 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108   × cxp 5683  dom cdm 5685  cfv 6561  Basecbs 17247  Posetcpo 18353  joincjn 18357  meetcmee 18358  Latclat 18476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-dm 5695  df-iota 6514  df-fv 6569  df-lat 18477
This theorem is referenced by:  odulatb  18479  latcl2  18481  latlem  18482  latpos  18483  latjcom  18492  latmcom  18508  clatl  18553  toslat  48871
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