Theorem isclat 18218

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 6774	. . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2		isclat.u	. . . . . 6 ⊢ 𝑈 = (lub‘𝐾)
3	1, 2	eqtr4di 2796	. . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
4	3	dmeqd 5814	. . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5		fveq2 6774	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		isclat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2796	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	pweqd 4552	. . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
9	4, 8	eqeq12d 2754	. . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10		fveq2 6774	. . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11		isclat.g	. . . . . 6 ⊢ 𝐺 = (glb‘𝐾)
12	10, 11	eqtr4di 2796	. . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
13	12	dmeqd 5814	. . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
14	13, 8	eqeq12d 2754	. . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
15	9, 14	anbi12d 631	. 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 18217	. 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
17	15, 16	elrab2 3627	1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  𝒫 cpw 4533  dom cdm 5589  ‘cfv 6433  Basecbs 16912  Posetcpo 18025  lubclub 18027  glbcglb 18028  CLatccla 18216

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-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-dm 5599  df-iota 6391  df-fv 6441  df-clat 18217

This theorem is referenced by:  clatpos  18219  clatlem  18220  clatlubcl2  18222  clatglbcl2  18224  oduclatb  18225  clatl  18226  xrsclat  31289  isclatd  46269