Theorem islat 18614

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.

Step	Hyp	Ref	Expression
1		fveq2 6888	. . . . . 6 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2		islat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
3	1, 2	eqtr4di 2813	. . . . 5 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = ∨ )
4	3	dmeqd 5887	. . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ )
5		fveq2 6888	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		islat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2813	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	sqxpeqd 5682	. . . 4 ⊢ (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵))
9	4, 8	eqeq12d 2776	. . 3 ⊢ (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∨ = (𝐵 × 𝐵)))
10		fveq2 6888	. . . . . 6 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11		islat.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
12	10, 11	eqtr4di 2813	. . . . 5 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = ∧ )
13	12	dmeqd 5887	. . . 4 ⊢ (𝑙 = 𝐾 → dom (meet‘𝑙) = dom ∧ )
14	13, 8	eqeq12d 2776	. . 3 ⊢ (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∧ = (𝐵 × 𝐵)))
15	9, 14	anbi12d 644	. 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 18613	. 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
17	15, 16	elrab2 3648	1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144   × cxp 5648  dom cdm 5650  ‘cfv 6541  Basecbs 17394  Posetcpo 18488  joincjn 18492  meetcmee 18493  Latclat 18612

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-dm 5660  df-iota 6496  df-fv 6549  df-lat 18613

This theorem is used by:  odulatb  18615  latcl2  18617  latlem  18618  latpos  18619  latjcom  18628  latmcom  18644  clatl  18689  toslat  50203