P. 186 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ipolt 18501	Strict order condition of the inclusion poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐹)    &   ⊢ < = (lt‘𝐼)    ⇒   ⊢ ((𝐹 ∈ 𝑉 ∧ 𝑋 ∈ 𝐹 ∧ 𝑌 ∈ 𝐹) → (𝑋 < 𝑌 ↔ 𝑋 ⊊ 𝑌))
Theorem	ipopos 18502	The inclusion poset on a family of sets is actually a poset. (Contributed by Stefan O'Rear, 30-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐹)    ⇒   ⊢ 𝐼 ∈ Poset
Theorem	isipodrs 18503*	Condition for a family of sets to be directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ((toInc‘𝐴) ∈ Dirset ↔ (𝐴 ∈ V ∧ 𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐴 (𝑥 ∪ 𝑦) ⊆ 𝑧))
Theorem	ipodrscl 18504	Direction by inclusion as used here implies sethood. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ((toInc‘𝐴) ∈ Dirset → 𝐴 ∈ V)
Theorem	ipodrsfi 18505*	Finite upper bound property for directed collections of sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (((toInc‘𝐴) ∈ Dirset ∧ 𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin) → ∃𝑧 ∈ 𝐴 ∪ 𝑋 ⊆ 𝑧)
Theorem	fpwipodrs 18506	The finite subsets of any set are directed by inclusion. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐴 ∈ 𝑉 → (toInc‘(𝒫 𝐴 ∩ Fin)) ∈ Dirset)
Theorem	ipodrsima 18507*	The monotone image of a directed set. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝜑 → 𝐹 Fn 𝒫 𝐴)    &   ⊢ ((𝜑 ∧ (𝑢 ⊆ 𝑣 ∧ 𝑣 ⊆ 𝐴)) → (𝐹‘𝑢) ⊆ (𝐹‘𝑣))    &   ⊢ (𝜑 → (toInc‘𝐵) ∈ Dirset)    &   ⊢ (𝜑 → 𝐵 ⊆ 𝒫 𝐴)    &   ⊢ (𝜑 → (𝐹 “ 𝐵) ∈ 𝑉)    ⇒   ⊢ (𝜑 → (toInc‘(𝐹 “ 𝐵)) ∈ Dirset)
Theorem	isacs3lem 18508*	An algebraic closure system satisfies isacs3 18516. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
Theorem	acsdrsel 18509	An algebraic closure system contains all directed unions of closed sets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌 ⊆ 𝐶 ∧ (toInc‘𝑌) ∈ Dirset) → ∪ 𝑌 ∈ 𝐶)
Theorem	isacs4lem 18510*	In a closure system in which directed unions of closed sets are closed, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))))
Theorem	isacs5lem 18511*	If closure commutes with directed unions, then the closure of a set is the closure of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑡 ∈ 𝒫 𝒫 𝑋((toInc‘𝑡) ∈ Dirset → (𝐹‘∪ 𝑡) = ∪ (𝐹 “ 𝑡))) → (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin))))
Theorem	acsdrscl 18512	In an algebraic closure system, closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑌 ⊆ 𝒫 𝑋 ∧ (toInc‘𝑌) ∈ Dirset) → (𝐹‘∪ 𝑌) = ∪ (𝐹 “ 𝑌))
Theorem	acsficl 18513	A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝐹‘𝑆) = ∪ (𝐹 “ (𝒫 𝑆 ∩ Fin)))
Theorem	isacs5 18514*	A closure system is algebraic iff the closure of a generating set is the union of the closures of its finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝐹‘𝑠) = ∪ (𝐹 “ (𝒫 𝑠 ∩ Fin))))
Theorem	isacs4 18515*	A closure system is algebraic iff closure commutes with directed unions. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ 𝐹 = (mrCls‘𝐶)    ⇒   ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝒫 𝑋((toInc‘𝑠) ∈ Dirset → (𝐹‘∪ 𝑠) = ∪ (𝐹 “ 𝑠))))
Theorem	isacs3 18516*	A closure system is algebraic iff directed unions of closed sets are closed. (Contributed by Stefan O'Rear, 2-Apr-2015.)
⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝐶((toInc‘𝑠) ∈ Dirset → ∪ 𝑠 ∈ 𝐶)))
Theorem	acsficld 18517	In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 18513. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑁‘𝑆) = ∪ (𝑁 “ (𝒫 𝑆 ∩ Fin)))
Theorem	acsficl2d 18518*	In an algebraic closure system, an element is in the closure of a set if and only if it is in the closure of a finite subset. Alternate form of acsficl 18513. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑌 ∈ (𝑁‘𝑆) ↔ ∃𝑥 ∈ (𝒫 𝑆 ∩ Fin)𝑌 ∈ (𝑁‘𝑥)))
Theorem	acsfiindd 18519	In an algebraic closure system, a set is independent if and only if all its finite subsets are independent. Part of Proposition 4.1.3 in [FaureFrolicher] p. 83. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    ⇒   ⊢ (𝜑 → (𝑆 ∈ 𝐼 ↔ (𝒫 𝑆 ∩ Fin) ⊆ 𝐼))
Theorem	acsmapd 18520*	In an algebraic closure system, if 𝑇 is contained in the closure of 𝑆, there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that the closure of ∪ ran 𝑓 contains 𝑇. This is proven by applying acsficl2d 18518 to each element of 𝑇. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ (𝜑 → 𝑆 ⊆ 𝑋)    &   ⊢ (𝜑 → 𝑇 ⊆ (𝑁‘𝑆))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑇 ⊆ (𝑁‘∪ ran 𝑓)))
Theorem	acsmap2d 18521*	In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is independent, then there is a map 𝑓 from 𝑇 into the set of finite subsets of 𝑆 such that 𝑆 equals the union of ran 𝑓. This is proven by taking the map 𝑓 from acsmapd 18520 and observing that, since 𝑆 and 𝑇 have the same closure, the closure of ∪ ran 𝑓 must contain 𝑆. Since 𝑆 is independent, by mrissmrcd 17608, ∪ ran 𝑓 must equal 𝑆. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    ⇒   ⊢ (𝜑 → ∃𝑓(𝑓:𝑇⟶(𝒫 𝑆 ∩ Fin) ∧ 𝑆 = ∪ ran 𝑓))
Theorem	acsinfd 18522	In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 is infinite. This follows from applying unirnffid 9305 to the map given in acsmap2d 18521. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    &   ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)    ⇒   ⊢ (𝜑 → ¬ 𝑇 ∈ Fin)
Theorem	acsdomd 18523	In an algebraic closure system, if 𝑆 and 𝑇 have the same closure and 𝑆 is infinite independent, then 𝑇 dominates 𝑆. This follows from applying acsinfd 18522 and then applying unirnfdomd 10527 to the map given in acsmap2d 18521. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ⊆ 𝑋)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    &   ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑆 ≼ 𝑇)
Theorem	acsinfdimd 18524	In an algebraic closure system, if two independent sets have equal closure and one is infinite, then they are equinumerous. This is proven by using acsdomd 18523 twice with acsinfd 18522. See Section II.5 in [Cohn] p. 81 to 82. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐼)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    &   ⊢ (𝜑 → ¬ 𝑆 ∈ Fin)    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
Theorem	acsexdimd 18525*	In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 17618 for the finite case and acsinfdimd 18524 for the infinite case. This is a special case of Theorem 4.2.2 in [FaureFrolicher] p. 87. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 ∈ (ACS‘𝑋))    &   ⊢ 𝑁 = (mrCls‘𝐴)    &   ⊢ 𝐼 = (mrInd‘𝐴)    &   ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧})))    &   ⊢ (𝜑 → 𝑆 ∈ 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝐼)    &   ⊢ (𝜑 → (𝑁‘𝑆) = (𝑁‘𝑇))    ⇒   ⊢ (𝜑 → 𝑆 ≈ 𝑇)
Theorem	mrelatglb 18526	Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatglbALT 48988 for an alternate proof.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
Theorem	mrelatglb0 18527	The empty intersection in a Moore space is realized by the base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → (𝐺‘∅) = 𝑋)
Theorem	mrelatlub 18528	Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatlubALT 48987 for an alternate proof.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐹 = (mrCls‘𝐶)    &   ⊢ 𝐿 = (lub‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
Theorem	mreclatBAD 18529*	A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7347 update): Reprove using isclat 18466 instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat 48989 for a good version.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ (𝐼 ∈ CLat ↔ (𝐼 ∈ Poset ∧ ∀𝑥(𝑥 ⊆ (Base‘𝐼) → (((lub‘𝐼)‘𝑥) ∈ (Base‘𝐼) ∧ ((glb‘𝐼)‘𝑥) ∈ (Base‘𝐼)))))    ⇒   ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐼 ∈ CLat)
9.6  Posets, directed sets, and lattices as relations
9.6.1  Posets and lattices as relations See commented-out notes for lattices as relations.
Syntax	cps 18530	Extend class notation with the class of all posets.
class PosetRel
Syntax	ctsr 18531	Extend class notation with the class of all totally ordered sets.
class TosetRel
Definition	df-ps 18532	Define the class of all posets (partially ordered sets) with weak ordering (e.g., "less than or equal to" instead of "less than"). A poset is a relation which is transitive, reflexive, and antisymmetric. (Contributed by NM, 11-May-2008.)
⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))}
Definition	df-tsr 18533	Define the class of all totally ordered sets. (Contributed by FL, 1-Nov-2009.)
⊢ TosetRel = {𝑟 ∈ PosetRel ∣ (dom 𝑟 × dom 𝑟) ⊆ (𝑟 ∪ ◡𝑟)}
Theorem	isps 18534	The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.)
⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))))
Theorem	psrel 18535	A poset is a relation. (Contributed by NM, 12-May-2008.)
⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
Theorem	psref2 18536	A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
Theorem	pstr2 18537	A poset is transitive. (Contributed by FL, 3-Aug-2009.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
Theorem	pslem 18538	Lemma for psref 18540 and others. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ (𝑅 ∈ PosetRel → (((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) ∧ (𝐴 ∈ ∪ ∪ 𝑅 → 𝐴𝑅𝐴) ∧ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)))
Theorem	psdmrn 18539	The domain and range of a poset equal its field. (Contributed by NM, 13-May-2008.)
⊢ (𝑅 ∈ PosetRel → (dom 𝑅 = ∪ ∪ 𝑅 ∧ ran 𝑅 = ∪ ∪ 𝑅))
Theorem	psref 18540	A poset is reflexive. (Contributed by NM, 13-May-2008.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
Theorem	psrn 18541	The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
Theorem	psasym 18542	A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)
Theorem	pstr 18543	A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	cnvps 18544	The converse of a poset is a poset. In the general case (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18545 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
Theorem	cnvpsb 18545	The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel))
Theorem	psss 18546	Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Theorem	psssdm2 18547	Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))
Theorem	psssdm 18548	Field of a subposet. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ⊆ 𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
Theorem	istsr 18549	The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ (𝑋 × 𝑋) ⊆ (𝑅 ∪ ◡𝑅)))
Theorem	istsr2 18550*	The predicate is a toset. (Contributed by FL, 1-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ TosetRel ↔ (𝑅 ∈ PosetRel ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑥)))
Theorem	tsrlin 18551	A toset is a linear order. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ TosetRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑅𝐵 ∨ 𝐵𝑅𝐴))
Theorem	tsrlemax 18552	Two ways of saying a number is less than or equal to the maximum of two others. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ TosetRel ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝐴𝑅if(𝐵𝑅𝐶, 𝐶, 𝐵) ↔ (𝐴𝑅𝐵 ∨ 𝐴𝑅𝐶)))
Theorem	tsrps 18553	A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Theorem	cnvtsr 18554	The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
Theorem	tsrss 18555	Any subset of a totally ordered set is totally ordered. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 21-Nov-2013.)
⊢ (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ TosetRel )
Theorem	ledm 18556	The domain of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
⊢ ℝ* = dom ≤
Theorem	lern 18557	The range of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ ℝ* = ran ≤
Theorem	lefld 18558	The field of the 'less or equal to' relationship on the extended real. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
⊢ ℝ* = ∪ ∪ ≤
Theorem	letsr 18559	The "less than or equal to" relationship on the extended reals is a toset. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ ≤ ∈ TosetRel
9.6.2  Directed sets, nets
Syntax	cdir 18560	Extend class notation with the class of directed sets.
class DirRel
Syntax	ctail 18561	Extend class notation with the tail function for directed sets.
class tail
Definition	df-dir 18562	Define the class of directed sets (the order relation itself is sometimes called a direction, and a directed set is a set equipped with a direction). (Contributed by Jeff Hankins, 25-Nov-2009.)
⊢ DirRel = {𝑟 ∣ ((Rel 𝑟 ∧ ( I ↾ ∪ ∪ 𝑟) ⊆ 𝑟) ∧ ((𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (∪ ∪ 𝑟 × ∪ ∪ 𝑟) ⊆ (◡𝑟 ∘ 𝑟)))}
Definition	df-tail 18563*	Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)
⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
Theorem	isdir 18564	A condition for a relation to be a direction. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝐴 = ∪ ∪ 𝑅    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝐴) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝐴 × 𝐴) ⊆ (◡𝑅 ∘ 𝑅)))))
Theorem	reldir 18565	A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (𝑅 ∈ DirRel → Rel 𝑅)
Theorem	dirdm 18566	A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
Theorem	dirref 18567	A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
Theorem	dirtr 18568	A direction is transitive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (((𝑅 ∈ DirRel ∧ 𝐶 ∈ 𝑉) ∧ (𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶)) → 𝐴𝑅𝐶)
Theorem	dirge 18569*	For any two elements of a directed set, there exists a third element greater than or equal to both. Note that this does not say that the two elements have a least upper bound. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝑅𝑥 ∧ 𝐵𝑅𝑥))
Theorem	tsrdir 18570	A totally ordered set is a directed set. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
⊢ (𝐴 ∈ TosetRel → 𝐴 ∈ DirRel)
PART 10  BASIC ALGEBRAIC STRUCTURES
10.1  Monoids
10.1.1  Magmas According to Wikipedia ("Magma (algebra)", 08-Jan-2020, https://en.wikipedia.org/wiki/magma_(algebra)) "In abstract algebra, a magma [...] is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation. The binary operation must be closed by definition but no other properties are imposed.". Since the concept of a "binary operation" is used in different variants, these differences are explained in more detail in the following: With df-mpo 7395, binary operations are defined by a rule, and with df-ov 7393, the value of a binary operation applied to two operands can be expressed. In both cases, the two operands can belong to different sets, and the result can be an element of a third set. However, according to Wikipedia "Binary operation", see https://en.wikipedia.org/wiki/Binary_operation 7393 (19-Jan-2020), "... a binary operation on a set 𝑆 is a mapping of the elements of the Cartesian product 𝑆 × 𝑆 to S: 𝑓:𝑆 × 𝑆⟶𝑆. Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S (or sometimes expressed as having the property of closure).". To distinguish this more restrictive definition (in Wikipedia and most of the literature) from the general case, binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 are more precisely called internal binary operations. If, in addition, the result is also contained in the set 𝑆, the operation should be called closed internal binary operation. Therefore, a "binary operation on a set 𝑆" according to Wikipedia is a "closed internal binary operation" in a more precise terminology. If the sets are different, the operation is explicitly called external binary operation (see Wikipedia https://en.wikipedia.org/wiki/Binary_operation#External_binary_operations 7393). The definition of magmas (Mgm, see df-mgm 18574) concentrates on the closure property of the associated operation, and poses no additional restrictions on it. In this way, it is most general and flexible.
Syntax	cplusf 18571	Extend class notation with group addition as a function.
class +𝑓
Syntax	cmgm 18572	Extend class notation with class of all magmas.
class Mgm
Definition	df-plusf 18573*	Define group addition function. Usually we will use +g directly instead of +𝑓, and they have the same behavior in most cases. The main advantage of +𝑓 for any magma is that it is a guaranteed function (mgmplusf 18584), while +g only has closure (mgmcl 18577). (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
Definition	df-mgm 18574*	A magma is a set equipped with an everywhere defined internal operation. Definition 1 in [BourbakiAlg1] p. 1, or definition of a groupoid in section I.1 of [Bruck] p. 1. Note: The term "groupoid" is now widely used to refer to other objects: (small) categories all of whose morphisms are invertible, or groups with a partial function replacing the binary operation. Therefore, we will only use the term "magma" for the present notion in set.mm. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
⊢ Mgm = {𝑔 ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏}
Theorem	ismgm 18575*	The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
Theorem	ismgmn0 18576*	The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵))
Theorem	mgmcl 18577	Closure of the operation of a magma. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 13-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mgm ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⚬ 𝑌) ∈ 𝐵)
Theorem	isnmgm 18578	A condition for a structure not to be a magma. (Contributed by AV, 30-Jan-2020.) (Proof shortened by NM, 5-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 ⚬ 𝑌) ∉ 𝐵) → 𝑀 ∉ Mgm)
Theorem	mgmsscl 18579	If the base set of a magma is contained in the base set of another magma, and the group operation of the magma is the restriction of the group operation of the other magma to its base set, then the base set of the magma is closed under the group operation of the other magma. Formerly part of proof of grpissubg 19085. (Contributed by AV, 17-Feb-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 𝑆 = (Base‘𝐻)    ⇒   ⊢ (((𝐺 ∈ Mgm ∧ 𝐻 ∈ Mgm) ∧ (𝑆 ⊆ 𝐵 ∧ (+g‘𝐻) = ((+g‘𝐺) ↾ (𝑆 × 𝑆))) ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆)) → (𝑋(+g‘𝐺)𝑌) ∈ 𝑆)
Theorem	plusffval 18580*	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Proof shortened by AV, 2-Mar-2024.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ⨣ = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 + 𝑦))
Theorem	plusfval 18581	The group addition operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ⨣ 𝑌) = (𝑋 + 𝑌))
Theorem	plusfeq 18582	If the addition operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ( + Fn (𝐵 × 𝐵) → ⨣ = + )
Theorem	plusffn 18583	The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ⨣ Fn (𝐵 × 𝐵)
Theorem	mgmplusf 18584	The group addition function of a magma is a function into its base set. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revisd by AV, 28-Jan-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⨣ = (+𝑓‘𝑀)    ⇒   ⊢ (𝑀 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
Theorem	mgmpropd 18585*	If two structures have the same (nonempty) base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, one is a magma iff the other one is. (Contributed by AV, 25-Feb-2020.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (𝐾 ∈ Mgm ↔ 𝐿 ∈ Mgm))
Theorem	ismgmd 18586*	Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → 𝐺 ∈ 𝑉)    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝐺 ∈ Mgm)
Theorem	issstrmgm 18587*	Characterize a substructure as submagma by closure properties. (Contributed by AV, 30-Aug-2021.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐻 = (𝐺 ↾s 𝑆)    ⇒   ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑆 ⊆ 𝐵) → (𝐻 ∈ Mgm ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
Theorem	intopsn 18588	The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20692. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Theorem	mgmb1mgm1 18589	The only magma with a base set consisting of one element is the trivial magma (at least if its operation is an internal binary operation). (Contributed by AV, 23-Jan-2020.) (Revised by AV, 7-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ + = (+g‘𝑀)    ⇒   ⊢ ((𝑀 ∈ Mgm ∧ 𝑍 ∈ 𝐵 ∧ + Fn (𝐵 × 𝐵)) → (𝐵 = {𝑍} ↔ + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Theorem	mgm0 18590	Any set with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
Theorem	mgm0b 18591	The structure with an empty base set and any group operation is a magma. (Contributed by AV, 28-Aug-2021.)
⊢ {⟨(Base‘ndx), ∅⟩, ⟨(+g‘ndx), 𝑂⟩} ∈ Mgm
Theorem	mgm1 18592	The structure with one element and the only closed internal operation for a singleton is a magma. (Contributed by AV, 10-Feb-2020.)
⊢ 𝑀 = {⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩}    ⇒   ⊢ (𝐼 ∈ 𝑉 → 𝑀 ∈ Mgm)
Theorem	opifismgm 18593*	A structure with a group addition operation expressed by a conditional operator is a magma if both values of the conditional operator are contained in the base set. (Contributed by AV, 9-Feb-2020.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ (+g‘𝑀) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝜓, 𝐶, 𝐷))    &   ⊢ (𝜑 → 𝐵 ≠ ∅)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐶 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐷 ∈ 𝐵)    ⇒   ⊢ (𝜑 → 𝑀 ∈ Mgm)
10.1.2  Identity elements According to Wikipedia ("Identity element", 7-Feb-2020, https://en.wikipedia.org/wiki/Identity_element): "In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it.". Or in more detail "... an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity." We concentrate on two-sided identities in the following. The existence of an identity (an identity is unique if it exists, see mgmidmo 18594) is an important property of monoids (see mndid 18678), and therefore also for groups (see grpid 18914), but also for magmas not required to be associative. Magmas with an identity element are called "unital magmas" (see Definition 2 in [BourbakiAlg1] p. 12) or, if the magmas are cancellative, "loops" (see definition in [Bruck] p. 15). In the context of extensible structures, the identity element (of any magma 𝑀) is defined as "group identity element" (0g‘𝑀), see df-0g 17411. Related theorems which are already valid for magmas are provided in the following.
Theorem	mgmidmo 18594*	A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)
⊢ ∃*𝑢 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑢 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑢) = 𝑥)
Theorem	grpidval 18595*	The value of the identity element of a group. (Contributed by NM, 20-Aug-2011.) (Revised by Mario Carneiro, 2-Oct-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ 0 = (℩𝑒(𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
Theorem	grpidpropd 18596*	If two structures have the same base set, and the values of their group (addition) operations are equal for all pairs of elements of the base set, they have the same identity element. (Contributed by Mario Carneiro, 27-Nov-2014.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
Theorem	fn0g 18597	The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 0g Fn V
Theorem	0g0 18598	The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
⊢ ∅ = (0g‘∅)
Theorem	ismgmid 18599*	The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    ⇒   ⊢ (𝜑 → ((𝑈 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑈 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑈) = 𝑥)) ↔ 0 = 𝑈))
Theorem	mgmidcl 18600*	The identity element of a magma, if it exists, belongs to the base set. (Contributed by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))    ⇒   ⊢ (𝜑 → 0 ∈ 𝐵)