Theorem isclat 18466

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 6861	. . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2		isclat.u	. . . . . 6 ⊢ 𝑈 = (lub‘𝐾)
3	1, 2	eqtr4di 2783	. . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
4	3	dmeqd 5872	. . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5		fveq2 6861	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		isclat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2783	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	pweqd 4583	. . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
9	4, 8	eqeq12d 2746	. . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10		fveq2 6861	. . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11		isclat.g	. . . . . 6 ⊢ 𝐺 = (glb‘𝐾)
12	10, 11	eqtr4di 2783	. . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
13	12	dmeqd 5872	. . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
14	13, 8	eqeq12d 2746	. . 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 18465	. 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
17	15, 16	elrab2 3665	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 4566  dom cdm 5641  ‘cfv 6514  Basecbs 17186  Posetcpo 18275  lubclub 18277  glbcglb 18278  CLatccla 18464

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 2702

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 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-dm 5651  df-iota 6467  df-fv 6522  df-clat 18465

This theorem is referenced by:  clatpos  18467  clatlem  18468  clatlubcl2  18470  clatglbcl2  18472  oduclatb  18473  clatl  18474  xrsclat  32956  isclatd  48975