Theorem islat 18151

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 6774	. . . . . 6 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2		islat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
3	1, 2	eqtr4di 2796	. . . . 5 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = ∨ )
4	3	dmeqd 5814	. . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ )
5		fveq2 6774	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		islat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2796	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	sqxpeqd 5621	. . . 4 ⊢ (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵))
9	4, 8	eqeq12d 2754	. . 3 ⊢ (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∨ = (𝐵 × 𝐵)))
10		fveq2 6774	. . . . . 6 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11		islat.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
12	10, 11	eqtr4di 2796	. . . . 5 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = ∧ )
13	12	dmeqd 5814	. . . 4 ⊢ (𝑙 = 𝐾 → dom (meet‘𝑙) = dom ∧ )
14	13, 8	eqeq12d 2754	. . 3 ⊢ (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∧ = (𝐵 × 𝐵)))
15	9, 14	anbi12d 631	. 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 18150	. 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
17	15, 16	elrab2 3627	1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   × cxp 5587  dom cdm 5589  ‘cfv 6433  Basecbs 16912  Posetcpo 18025  joincjn 18029  meetcmee 18030  Latclat 18149

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-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599  df-iota 6391  df-fv 6441  df-lat 18150

This theorem is referenced by:  odulatb  18152  latcl2  18154  latlem  18155  latpos  18156  latjcom  18165  latmcom  18181  clatl  18226  toslat  46268