Theorem islat 17773

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 6674	. . . . . 6 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2		islat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
3	1, 2	eqtr4di 2791	. . . . 5 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = ∨ )
4	3	dmeqd 5748	. . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ )
5		fveq2 6674	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		islat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	eqtr4di 2791	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	sqxpeqd 5557	. . . 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 6674	. . . . . 6 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11		islat.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
12	10, 11	eqtr4di 2791	. . . . 5 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = ∧ )
13	12	dmeqd 5748	. . . 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 634	. 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 17772	. 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
17	15, 16	elrab2 3591	1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1542   ∈ wcel 2114   × cxp 5523  dom cdm 5525  ‘cfv 6339  Basecbs 16586  Posetcpo 17666  joincjn 17670  meetcmee 17671  Latclat 17771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-rab 3062  df-v 3400  df-un 3848  df-in 3850  df-ss 3860  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-xp 5531  df-dm 5535  df-iota 6297  df-fv 6347  df-lat 17772

This theorem is referenced by:  latcl2  17774  latlem  17775  latpos  17776  latjcom  17785  latmcom  17801  clatl  17842  odulatb  17869