Theorem islat 18521

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 6878	. . . . . 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 5889	. . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ )
5		fveq2 6878	. . . . . 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 5687	. . . 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 6878	. . . . . 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 5889	. . . 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 18520	. 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
17	15, 16	elrab2 3649	1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   × cxp 5653  dom cdm 5655  ‘cfv 6533  Basecbs 17301  Posetcpo 18395  joincjn 18399  meetcmee 18400  Latclat 18519

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-dm 5665  df-iota 6489  df-fv 6541  df-lat 18520

This theorem is used by:  odulatb  18522  latcl2  18524  latlem  18525  latpos  18526  latjcom  18535  latmcom  18551  clatl  18596  toslat  49908