P. 184 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18301-18400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	join0 18301	Lemma for odumeet 18306. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ (join‘∅) = ∅
Theorem	meet0 18302	Lemma for odujoin 18304. (Contributed by Stefan O'Rear, 29-Jan-2015.) TODO (df-riota 7298 update): This proof increased from 152 bytes to 547 bytes after the df-riota 7298 change. Any way to shorten it? join0 18301 also.
⊢ (meet‘∅) = ∅
Theorem	odulub 18303	Least upper bounds in a dual order are greatest lower bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐷 = (ODual‘𝑂)    &   ⊢ 𝐿 = (glb‘𝑂)    ⇒   ⊢ (𝑂 ∈ 𝑉 → 𝐿 = (lub‘𝐷))
Theorem	odujoin 18304	Joins in a dual order are meets in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐷 = (ODual‘𝑂)    &   ⊢ ∧ = (meet‘𝑂)    ⇒   ⊢ ∧ = (join‘𝐷)
Theorem	oduglb 18305	Greatest lower bounds in a dual order are least upper bounds in the original order. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐷 = (ODual‘𝑂)    &   ⊢ 𝑈 = (lub‘𝑂)    ⇒   ⊢ (𝑂 ∈ 𝑉 → 𝑈 = (glb‘𝐷))
Theorem	odumeet 18306	Meets in a dual order are joins in the original. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐷 = (ODual‘𝑂)    &   ⊢ ∨ = (join‘𝑂)    ⇒   ⊢ ∨ = (meet‘𝐷)
Theorem	poslubmo 18307*	Least upper bounds in a poset are unique if they exist. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by NM, 16-Jun-2017.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐵 = (Base‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
Theorem	posglbmo 18308*	Greatest lower bounds in a poset are unique if they exist. (Contributed by NM, 20-Sep-2018.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐵 = (Base‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Poset ∧ 𝑆 ⊆ 𝐵) → ∃*𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
Theorem	poslubd 18309*	Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)
Theorem	poslubdg 18310*	Properties which determine the least upper bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝑈 = (lub‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ≤ 𝑇)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑥 ≤ 𝑦) → 𝑇 ≤ 𝑦)    ⇒   ⊢ (𝜑 → (𝑈‘𝑆) = 𝑇)
Theorem	posglbdg 18311*	Properties which determine the greatest lower bound in a poset. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐺 = (glb‘𝐾))    &   ⊢ (𝜑 → 𝐾 ∈ Poset)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑇 ≤ 𝑥)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝑆 𝑦 ≤ 𝑥) → 𝑦 ≤ 𝑇)    ⇒   ⊢ (𝜑 → (𝐺‘𝑆) = 𝑇)
9.4  Totally ordered sets (tosets)
Syntax	ctos 18312	Extend class notation with the class of all tosets.
class Toset
Definition	df-toset 18313*	Define the class of totally ordered sets (tosets). (Contributed by FL, 17-Nov-2014.)
⊢ Toset = {𝑓 ∈ Poset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑟𝑦 ∨ 𝑦𝑟𝑥)}
Theorem	istos 18314*	The predicate "is a toset". (Contributed by FL, 17-Nov-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ (𝐾 ∈ Toset ↔ (𝐾 ∈ Poset ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 ∨ 𝑦 ≤ 𝑥)))
Theorem	tosso 18315	Write the totally ordered set structure predicate in terms of the proper class strict order predicate. (Contributed by Mario Carneiro, 8-Feb-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ Toset ↔ ( < Or 𝐵 ∧ ( I ↾ 𝐵) ⊆ ≤ )))
Theorem	tospos 18316	A Toset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ (𝐹 ∈ Toset → 𝐹 ∈ Poset)
Theorem	tleile 18317	In a Toset, any two elements are comparable. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
Theorem	tltnle 18318	In a Toset, "less than" is equivalent to the negation of the converse of "less than or equal to", see pltnle 18234. (Contributed by Thierry Arnoux, 11-Feb-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))
Syntax	cp0 18319	Extend class notation with poset zero.
class 0.
Syntax	cp1 18320	Extend class notation with poset unit.
class 1.
Definition	df-p0 18321	Define poset zero. (Contributed by NM, 12-Oct-2011.)
⊢ 0. = (𝑝 ∈ V ↦ ((glb‘𝑝)‘(Base‘𝑝)))
Definition	df-p1 18322	Define poset unit. (Contributed by NM, 22-Oct-2011.)
⊢ 1. = (𝑝 ∈ V ↦ ((lub‘𝑝)‘(Base‘𝑝)))
Theorem	p0val 18323	Value of poset zero. (Contributed by NM, 12-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → 0 = (𝐺‘𝐵))
Theorem	p1val 18324	Value of poset zero. (Contributed by NM, 22-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → 1 = (𝑈‘𝐵))
Theorem	p0le 18325	Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝐺)    ⇒   ⊢ (𝜑 → 0 ≤ 𝑋)
Theorem	ple1 18326	Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ dom 𝑈)    ⇒   ⊢ (𝜑 → 𝑋 ≤ 1 )
Theorem	resspos 18327	The restriction of a Poset is a Poset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ ((𝐹 ∈ Poset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Poset)
Theorem	resstos 18328	The restriction of a Toset is a Toset. (Contributed by Thierry Arnoux, 20-Jan-2018.)
⊢ ((𝐹 ∈ Toset ∧ 𝐴 ∈ 𝑉) → (𝐹 ↾s 𝐴) ∈ Toset)
9.5  Lattices
9.5.1  Lattices
Syntax	clat 18329	Extend class notation with the class of all lattices.
class Lat
Definition	df-lat 18330	Define the class of all lattices. A lattice is a poset in which the join and meet of any two elements always exists. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
⊢ Lat = {𝑝 ∈ Poset ∣ (dom (join‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)) ∧ dom (meet‘𝑝) = ((Base‘𝑝) × (Base‘𝑝)))}
Theorem	islat 18331	The predicate "is a lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (𝐾 ∈ Lat ↔ (𝐾 ∈ Poset ∧ (dom ∨ = (𝐵 × 𝐵) ∧ dom ∧ = (𝐵 × 𝐵))))
Theorem	odulatb 18332	Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐷 = (ODual‘𝑂)    ⇒   ⊢ (𝑂 ∈ 𝑉 → (𝑂 ∈ Lat ↔ 𝐷 ∈ Lat))
Theorem	odulat 18333	Being a lattice is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐷 = (ODual‘𝑂)    ⇒   ⊢ (𝑂 ∈ Lat → 𝐷 ∈ Lat)
Theorem	latcl2 18334	The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Lat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ ∧ ⟨𝑋, 𝑌⟩ ∈ dom ∧ ))
Theorem	latlem 18335	Lemma for lattice properties. (Contributed by NM, 14-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ∨ 𝑌) ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ∈ 𝐵))
Theorem	latpos 18336	A lattice is a poset. (Contributed by NM, 17-Sep-2011.)
⊢ (𝐾 ∈ Lat → 𝐾 ∈ Poset)
Theorem	latjcl 18337	Closure of join operation in a lattice. (chjcom 31476 analog.) (Contributed by NM, 14-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵)
Theorem	latmcl 18338	Closure of meet operation in a lattice. (incom 4157 analog.) (Contributed by NM, 14-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ∈ 𝐵)
Theorem	latref 18339	A lattice ordering is reflexive. (ssid 3955 analog.) (Contributed by NM, 8-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → 𝑋 ≤ 𝑋)
Theorem	latasymb 18340	A lattice ordering is asymmetric. (eqss 3948 analog.) (Contributed by NM, 22-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))
Theorem	latasym 18341	A lattice ordering is asymmetric. (eqss 3948 analog.) (Contributed by NM, 8-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))
Theorem	lattr 18342	A lattice ordering is transitive. (sstr 3941 analog.) (Contributed by NM, 17-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 ≤ 𝑍))
Theorem	latasymd 18343	Deduce equality from lattice ordering. (eqssd 3950 analog.) (Contributed by NM, 18-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Lat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑋)    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	lattrd 18344	A lattice ordering is transitive. Deduction version of lattr 18342. (Contributed by NM, 3-Sep-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ (𝜑 → 𝐾 ∈ Lat)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≤ 𝑌)    &   ⊢ (𝜑 → 𝑌 ≤ 𝑍)    ⇒   ⊢ (𝜑 → 𝑋 ≤ 𝑍)
Theorem	latjcom 18345	The join of a lattice commutes. (chjcom 31476 analog.) (Contributed by NM, 16-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋))
Theorem	latlej1 18346	A join's first argument is less than or equal to the join. (chub1 31477 analog.) (Contributed by NM, 17-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑋 ≤ (𝑋 ∨ 𝑌))
Theorem	latlej2 18347	A join's second argument is less than or equal to the join. (chub2 31478 analog.) (Contributed by NM, 17-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝑌 ≤ (𝑋 ∨ 𝑌))
Theorem	latjle12 18348	A join is less than or equal to a third value iff each argument is less than or equal to the third value. (chlub 31479 analog.) (Contributed by NM, 17-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ 𝑌 ≤ 𝑍) ↔ (𝑋 ∨ 𝑌) ≤ 𝑍))
Theorem	latleeqj1 18349	"Less than or equal to" in terms of join. (chlejb1 31482 analog.) (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 ∨ 𝑌) = 𝑌))
Theorem	latleeqj2 18350	"Less than or equal to" in terms of join. (chlejb2 31483 analog.) (Contributed by NM, 14-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑌 ∨ 𝑋) = 𝑌))
Theorem	latjlej1 18351	Add join to both sides of a lattice ordering. (chlej1i 31443 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑍)))
Theorem	latjlej2 18352	Add join to both sides of a lattice ordering. (chlej2i 31444 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑍 ∨ 𝑋) ≤ (𝑍 ∨ 𝑌)))
Theorem	latjlej12 18353	Add join to both sides of a lattice ordering. (chlej12i 31445 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑍 ≤ 𝑊) → (𝑋 ∨ 𝑍) ≤ (𝑌 ∨ 𝑊)))
Theorem	latnlej 18354	An idiom to express that a lattice element differs from two others. (Contributed by NM, 28-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → (𝑋 ≠ 𝑌 ∧ 𝑋 ≠ 𝑍))
Theorem	latnlej1l 18355	An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → 𝑋 ≠ 𝑌)
Theorem	latnlej1r 18356	An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → 𝑋 ≠ 𝑍)
Theorem	latnlej2 18357	An idiom to express that a lattice element differs from two others. (Contributed by NM, 10-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → (¬ 𝑋 ≤ 𝑌 ∧ ¬ 𝑋 ≤ 𝑍))
Theorem	latnlej2l 18358	An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → ¬ 𝑋 ≤ 𝑌)
Theorem	latnlej2r 18359	An idiom to express that a lattice element differs from two others. (Contributed by NM, 19-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ ¬ 𝑋 ≤ (𝑌 ∨ 𝑍)) → ¬ 𝑋 ≤ 𝑍)
Theorem	latjidm 18360	Lattice join is idempotent. Analogue of unidm 4105. (Contributed by NM, 8-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∨ 𝑋) = 𝑋)
Theorem	latmcom 18361	The join of a lattice commutes. (Contributed by NM, 6-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) = (𝑌 ∧ 𝑋))
Theorem	latmle1 18362	A meet is less than or equal to its first argument. (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑋)
Theorem	latmle2 18363	A meet is less than or equal to its second argument. (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ 𝑌) ≤ 𝑌)
Theorem	latlem12 18364	An element is less than or equal to a meet iff the element is less than or equal to each argument of the meet. (Contributed by NM, 21-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≤ 𝑍) ↔ 𝑋 ≤ (𝑌 ∧ 𝑍)))
Theorem	latleeqm1 18365	"Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) = 𝑋))
Theorem	latleeqm2 18366	"Less than or equal to" in terms of meet. (Contributed by NM, 7-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑌 ∧ 𝑋) = 𝑋))
Theorem	latmlem1 18367	Add meet to both sides of a lattice ordering. (Contributed by NM, 10-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑍)))
Theorem	latmlem2 18368	Add meet to both sides of a lattice ordering. (sslin 4191 analog.) (Contributed by NM, 10-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑍 ∧ 𝑋) ≤ (𝑍 ∧ 𝑌)))
Theorem	latmlem12 18369	Add join to both sides of a lattice ordering. (ss2in 4193 analog.) (Contributed by NM, 10-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑍 ≤ 𝑊) → (𝑋 ∧ 𝑍) ≤ (𝑌 ∧ 𝑊)))
Theorem	latnlemlt 18370	Negation of "less than or equal to" expressed in terms of meet and less-than. (nssinpss 4215 analog.) (Contributed by NM, 5-Feb-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑋 ≤ 𝑌 ↔ (𝑋 ∧ 𝑌) < 𝑋))
Theorem	latnle 18371	Equivalent expressions for "not less than" in a lattice. (chnle 31484 analog.) (Contributed by NM, 16-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (¬ 𝑌 ≤ 𝑋 ↔ 𝑋 < (𝑋 ∨ 𝑌)))
Theorem	latmidm 18372	Lattice meet is idempotent. Analogue of inidm 4175. (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑋 ∧ 𝑋) = 𝑋)
Theorem	latabs1 18373	Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs1 31486 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ (𝑋 ∧ 𝑌)) = 𝑋)
Theorem	latabs2 18374	Lattice absorption law. From definition of lattice in [Kalmbach] p. 14. (chabs2 31487 analog.) (Contributed by NM, 8-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∧ (𝑋 ∨ 𝑌)) = 𝑋)
Theorem	latledi 18375	An ortholattice is distributive in one ordering direction. (ledi 31510 analog.) (Contributed by NM, 7-Nov-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍)) ≤ (𝑋 ∧ (𝑌 ∨ 𝑍)))
Theorem	latmlej11 18376	Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) ≤ (𝑋 ∨ 𝑍))
Theorem	latmlej12 18377	Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ 𝑌) ≤ (𝑍 ∨ 𝑋))
Theorem	latmlej21 18378	Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∧ 𝑋) ≤ (𝑋 ∨ 𝑍))
Theorem	latmlej22 18379	Ordering of a meet and join with a common variable. (Contributed by NM, 4-Oct-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ∧ 𝑋) ≤ (𝑍 ∨ 𝑋))
Theorem	lubsn 18380	The least upper bound of a singleton. (chsupsn 31383 analog.) (Contributed by NM, 20-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑈‘{𝑋}) = 𝑋)
Theorem	latjass 18381	Lattice join is associative. Lemma 2.2 in [MegPav2002] p. 362. (chjass 31503 analog.) (Contributed by NM, 17-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = (𝑋 ∨ (𝑌 ∨ 𝑍)))
Theorem	latj12 18382	Swap 1st and 2nd members of lattice join. (chj12 31504 analog.) (Contributed by NM, 4-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∨ 𝑍)) = (𝑌 ∨ (𝑋 ∨ 𝑍)))
Theorem	latj32 18383	Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 2-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ 𝑌))
Theorem	latj13 18384	Swap 1st and 3rd members of lattice join. (Contributed by NM, 4-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∨ 𝑍)) = (𝑍 ∨ (𝑌 ∨ 𝑋)))
Theorem	latj31 18385	Swap 2nd and 3rd members of lattice join. Lemma 2.2 in [MegPav2002] p. 362. (Contributed by NM, 23-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑍 ∨ 𝑌) ∨ 𝑋))
Theorem	latjrot 18386	Rotate lattice join of 3 classes. (Contributed by NM, 23-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑍 ∨ 𝑋) ∨ 𝑌))
Theorem	latj4 18387	Rearrangement of lattice join of 4 classes. (chj4 31505 analog.) (Contributed by NM, 14-Jun-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ (𝑍 ∨ 𝑊)) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑊)))
Theorem	latj4rot 18388	Rotate lattice join of 4 classes. (Contributed by NM, 11-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ (𝑍 ∨ 𝑊)) = ((𝑊 ∨ 𝑋) ∨ (𝑌 ∨ 𝑍)))
Theorem	latjjdi 18389	Lattice join distributes over itself. (Contributed by NM, 30-Jul-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∨ 𝑍)) = ((𝑋 ∨ 𝑌) ∨ (𝑋 ∨ 𝑍)))
Theorem	latjjdir 18390	Lattice join distributes over itself. (Contributed by NM, 2-Aug-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∨ 𝑌) ∨ 𝑍) = ((𝑋 ∨ 𝑍) ∨ (𝑌 ∨ 𝑍)))
Theorem	mod1ile 18391	The weak direction of the modular law (e.g., pmod1i 39866, atmod1i1 39875) that holds in any lattice. (Contributed by NM, 11-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑍 → (𝑋 ∨ (𝑌 ∧ 𝑍)) ≤ ((𝑋 ∨ 𝑌) ∧ 𝑍)))
Theorem	mod2ile 18392	The weak direction of the modular law (e.g., pmod2iN 39867) that holds in any lattice. (Contributed by NM, 11-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑍 ≤ 𝑋 → ((𝑋 ∧ 𝑌) ∨ 𝑍) ≤ (𝑋 ∧ (𝑌 ∨ 𝑍))))
Theorem	latmass 18393	Lattice meet is associative. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ∧ 𝑌) ∧ 𝑍) = (𝑋 ∧ (𝑌 ∧ 𝑍)))
Theorem	latdisdlem 18394*	Lemma for latdisd 18395. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (𝐾 ∈ Lat → (∀𝑢 ∈ 𝐵 ∀𝑣 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑢 ∨ (𝑣 ∧ 𝑤)) = ((𝑢 ∨ 𝑣) ∧ (𝑢 ∨ 𝑤)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
Theorem	latdisd 18395*	In a lattice, joins distribute over meets if and only if meets distribute over joins; the distributive property is self-dual. (Contributed by Stefan O'Rear, 29-Jan-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (𝐾 ∈ Lat → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∨ (𝑦 ∧ 𝑧)) = ((𝑥 ∨ 𝑦) ∧ (𝑥 ∨ 𝑧)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧))))
9.5.2  Complete lattices
Syntax	ccla 18396	Extend class notation with complete lattices.
class CLat
Definition	df-clat 18397	Define the class of all complete lattices, where every subset of the base set has an LUB and a GLB. (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
⊢ CLat = {𝑝 ∈ Poset ∣ (dom (lub‘𝑝) = 𝒫 (Base‘𝑝) ∧ dom (glb‘𝑝) = 𝒫 (Base‘𝑝))}
Theorem	isclat 18398	The predicate "is a complete lattice". (Contributed by NM, 18-Oct-2012.) (Revised by NM, 12-Sep-2018.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ (𝐾 ∈ CLat ↔ (𝐾 ∈ Poset ∧ (dom 𝑈 = 𝒫 𝐵 ∧ dom 𝐺 = 𝒫 𝐵)))
Theorem	clatpos 18399	A complete lattice is a poset. (Contributed by NM, 8-Sep-2018.)
⊢ (𝐾 ∈ CLat → 𝐾 ∈ Poset)
Theorem	clatlem 18400	Lemma for properties of a complete lattice. (Contributed by NM, 14-Sep-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑈 = (lub‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    ⇒   ⊢ ((𝐾 ∈ CLat ∧ 𝑆 ⊆ 𝐵) → ((𝑈‘𝑆) ∈ 𝐵 ∧ (𝐺‘𝑆) ∈ 𝐵))