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Theorem islat 18151
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 6774 . . . . . 6 (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2 islat.j . . . . . 6 = (join‘𝐾)
31, 2eqtr4di 2796 . . . . 5 (𝑙 = 𝐾 → (join‘𝑙) = )
43dmeqd 5814 . . . 4 (𝑙 = 𝐾 → dom (join‘𝑙) = dom )
5 fveq2 6774 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 islat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2796 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87sqxpeqd 5621 . . . 4 (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵))
94, 8eqeq12d 2754 . . 3 (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom = (𝐵 × 𝐵)))
10 fveq2 6774 . . . . . 6 (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11 islat.m . . . . . 6 = (meet‘𝐾)
1210, 11eqtr4di 2796 . . . . 5 (𝑙 = 𝐾 → (meet‘𝑙) = )
1312dmeqd 5814 . . . 4 (𝑙 = 𝐾 → dom (meet‘𝑙) = dom )
1413, 8eqeq12d 2754 . . 3 (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom = (𝐵 × 𝐵)))
159, 14anbi12d 631 . 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 18150 . 2 Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
1715, 16elrab2 3627 1 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wcel 2106   × cxp 5587  dom cdm 5589  cfv 6433  Basecbs 16912  Posetcpo 18025  joincjn 18029  meetcmee 18030  Latclat 18149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599  df-iota 6391  df-fv 6441  df-lat 18150
This theorem is referenced by:  odulatb  18152  latcl2  18154  latlem  18155  latpos  18156  latjcom  18165  latmcom  18181  clatl  18226  toslat  46268
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