Theorem isclat 17310

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 6352	. . . . . 6 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = (lub‘𝐾))
2		isclat.u	. . . . . 6 ⊢ 𝑈 = (lub‘𝐾)
3	1, 2	syl6eqr 2812	. . . . 5 ⊢ (𝑙 = 𝐾 → (lub‘𝑙) = 𝑈)
4	3	dmeqd 5481	. . . 4 ⊢ (𝑙 = 𝐾 → dom (lub‘𝑙) = dom 𝑈)
5		fveq2 6352	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		isclat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	syl6eqr 2812	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	pweqd 4307	. . . 4 ⊢ (𝑙 = 𝐾 → 𝒫 (Base‘𝑙) = 𝒫 𝐵)
9	4, 8	eqeq12d 2775	. . 3 ⊢ (𝑙 = 𝐾 → (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝑈 = 𝒫 𝐵))
10		fveq2 6352	. . . . . 6 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = (glb‘𝐾))
11		isclat.g	. . . . . 6 ⊢ 𝐺 = (glb‘𝐾)
12	10, 11	syl6eqr 2812	. . . . 5 ⊢ (𝑙 = 𝐾 → (glb‘𝑙) = 𝐺)
13	12	dmeqd 5481	. . . 4 ⊢ (𝑙 = 𝐾 → dom (glb‘𝑙) = dom 𝐺)
14	13, 8	eqeq12d 2775	. . 3 ⊢ (𝑙 = 𝐾 → (dom (glb‘𝑙) = 𝒫 (Base‘𝑙) ↔ dom 𝐺 = 𝒫 𝐵))
15	9, 14	anbi12d 749	. 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 17309	. 2 ⊢ CLat = {𝑙 ∈ Poset ∣ (dom (lub‘𝑙) = 𝒫 (Base‘𝑙) ∧ dom (glb‘𝑙) = 𝒫 (Base‘𝑙))}
17	15, 16	elrab2 3507	1 ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))

Syntax hints:   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  𝒫 cpw 4302  dom cdm 5266  ‘cfv 6049  Basecbs 16059  Posetcpo 17141  lubclub 17143  glbcglb 17144  CLatccla 17308

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-dm 5276  df-iota 6012  df-fv 6057  df-clat 17309

This theorem is referenced by:  clatpos  17311  clatlem  17312  clatlubcl2  17314  clatglbcl2  17316  clatl  17317  oduclatb  17345  xrsclat  29989