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Theorem islat 16968
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 6148 . . . . . 6 (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2 islat.j . . . . . 6 = (join‘𝐾)
31, 2syl6eqr 2673 . . . . 5 (𝑙 = 𝐾 → (join‘𝑙) = )
43dmeqd 5286 . . . 4 (𝑙 = 𝐾 → dom (join‘𝑙) = dom )
5 fveq2 6148 . . . . . 6 (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6 islat.b . . . . . 6 𝐵 = (Base‘𝐾)
75, 6syl6eqr 2673 . . . . 5 (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
87sqxpeqd 5101 . . . 4 (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵))
94, 8eqeq12d 2636 . . 3 (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom = (𝐵 × 𝐵)))
10 fveq2 6148 . . . . . 6 (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11 islat.m . . . . . 6 = (meet‘𝐾)
1210, 11syl6eqr 2673 . . . . 5 (𝑙 = 𝐾 → (meet‘𝑙) = )
1312dmeqd 5286 . . . 4 (𝑙 = 𝐾 → dom (meet‘𝑙) = dom )
1413, 8eqeq12d 2636 . . 3 (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom = (𝐵 × 𝐵)))
159, 14anbi12d 746 . 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 16967 . 2 Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
1715, 16elrab2 3348 1 (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom = (𝐵 × 𝐵) ∧ dom = (𝐵 × 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987   × cxp 5072  dom cdm 5074  cfv 5847  Basecbs 15781  Posetcpo 16861  joincjn 16865  meetcmee 16866  Latclat 16966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-dm 5084  df-iota 5810  df-fv 5855  df-lat 16967
This theorem is referenced by:  latcl2  16969  latlem  16970  latpos  16971  latjcom  16980  latmcom  16996  clatl  17037  odulatb  17064
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