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Theorem islat 17773
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 6674 . . . . . 6 (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2 islat.j . . . . . 6 = (join‘𝐾)
31, 2eqtr4di 2791 . . . . 5 (𝑙 = 𝐾 → (join‘𝑙) = )
43dmeqd 5748 . . . 4 (𝑙 = 𝐾 → dom (join‘𝑙) = dom )
5 fveq2 6674 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 islat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6eqtr4di 2791 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87sqxpeqd 5557 . . . 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 6674 . . . . . 6 (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11 islat.m . . . . . 6 = (meet‘𝐾)
1210, 11eqtr4di 2791 . . . . 5 (𝑙 = 𝐾 → (meet‘𝑙) = )
1312dmeqd 5748 . . . 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 634 . 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 17772 . 2 Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
1715, 16elrab2 3591 1 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wcel 2114   × cxp 5523  dom cdm 5525  cfv 6339  Basecbs 16586  Posetcpo 17666  joincjn 17670  meetcmee 17671  Latclat 17771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-dm 5535  df-iota 6297  df-fv 6347  df-lat 17772
This theorem is referenced by:  latcl2  17774  latlem  17775  latpos  17776  latjcom  17785  latmcom  17801  clatl  17842  odulatb  17869
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