Theorem islat 17659

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 6672	. . . . . 6 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = (join‘𝐾))
2		islat.j	. . . . . 6 ⊢ ∨ = (join‘𝐾)
3	1, 2	syl6eqr 2876	. . . . 5 ⊢ (𝑙 = 𝐾 → (join‘𝑙) = ∨ )
4	3	dmeqd 5776	. . . 4 ⊢ (𝑙 = 𝐾 → dom (join‘𝑙) = dom ∨ )
5		fveq2 6672	. . . . . 6 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = (Base‘𝐾))
6		islat.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
7	5, 6	syl6eqr 2876	. . . . 5 ⊢ (𝑙 = 𝐾 → (Base‘𝑙) = 𝐵)
8	7	sqxpeqd 5589	. . . 4 ⊢ (𝑙 = 𝐾 → ((Base‘𝑙) × (Base‘𝑙)) = (𝐵 × 𝐵))
9	4, 8	eqeq12d 2839	. . 3 ⊢ (𝑙 = 𝐾 → (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∨ = (𝐵 × 𝐵)))
10		fveq2 6672	. . . . . 6 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = (meet‘𝐾))
11		islat.m	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
12	10, 11	syl6eqr 2876	. . . . 5 ⊢ (𝑙 = 𝐾 → (meet‘𝑙) = ∧ )
13	12	dmeqd 5776	. . . 4 ⊢ (𝑙 = 𝐾 → dom (meet‘𝑙) = dom ∧ )
14	13, 8	eqeq12d 2839	. . 3 ⊢ (𝑙 = 𝐾 → (dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ↔ dom ∧ = (𝐵 × 𝐵)))
15	9, 14	anbi12d 632	. 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 17658	. 2 ⊢ Lat = {𝑙 ∈ Poset ∣ (dom (join‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)) ∧ dom (meet‘𝑙) = ((Base‘𝑙) × (Base‘𝑙)))}
17	15, 16	elrab2 3685	1 ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   × cxp 5555  dom cdm 5557  ‘cfv 6357  Basecbs 16485  Posetcpo 17552  joincjn 17556  meetcmee 17557  Latclat 17657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-xp 5563  df-dm 5567  df-iota 6316  df-fv 6365  df-lat 17658

This theorem is referenced by:  latcl2  17660  latlem  17661  latpos  17662  latjcom  17671  latmcom  17687  clatl  17728  odulatb  17755