Theorem isclat 18459

Description: The predicate "is a complete lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)

Hypotheses

Ref	Expression
isclat.b	⊢ 𝐵 = (Base‘𝐾)
isclat.u	⊢ 𝑈 = (lub‘𝐾)
isclat.g	⊢ 𝐺 = (glb‘𝐾)

Assertion

Ref	Expression
isclat	⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Proof of Theorem isclat

Dummy variable 𝑙 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2		isclat.u	. . . . . 6 ⊢ 𝑈 = (lub‘𝐾)
3	1, 2	eqtr4di 2782	. . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
4	3	dmeqd 5869	. . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5		fveq2 6858	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		isclat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2782	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	pweqd 4580	. . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
9	4, 8	eqeq12d 2745	. . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10		fveq2 6858	. . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11		isclat.g	. . . . . 6 ⊢ 𝐺 = (glb‘𝐾)
12	10, 11	eqtr4di 2782	. . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
13	12	dmeqd 5869	. . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
14	13, 8	eqeq12d 2745	. . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
15	9, 14	anbi12d 632	. 2 ⊢ (𝑙 = 𝐾 → ((dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙)) ↔ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
16		df-clat 18458	. 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
17	15, 16	elrab2 3662	1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  𝒫 cpw 4563  dom cdm 5638  ‘cfv 6511  Basecbs 17179  Posetcpo 18268  lubclub 18270  glbcglb 18271  CLatccla 18457

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-dm 5648  df-iota 6464  df-fv 6519  df-clat 18458

This theorem is referenced by:  clatpos  18460  clatlem  18461  clatlubcl2  18463  clatglbcl2  18465  oduclatb  18466  clatl  18467  xrsclat  32949  isclatd  48971