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	isacs4lem 18501*	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 18502*	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 18503	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 18504	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 18505*	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 18506*	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 18507*	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 18508	In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 18504. (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 18509*	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 18504. 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 18510	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 18511*	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 18509 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 18512*	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 18511 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 17585, ∪ 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 18513	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 9341 to the map given in acsmap2d 18512. 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 18514	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 18513 and then applying unirnfdomd 10559 to the map given in acsmap2d 18512. 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 18515	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 18514 twice with acsinfd 18513. 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 18516*	In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 17595 for the finite case and acsinfdimd 18515 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 18517	Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatglbALT 47833 for an alternate proof.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐺 = (glb‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈)
Theorem	mrelatglb0 18518	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 18519	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 47832 for an alternate proof.
⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ 𝐹 = (mrCls‘𝐶)    &   ⊢ 𝐿 = (lub‘𝐼)    ⇒   ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈))
Theorem	mreclatBAD 18520*	A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7358 update): Reprove using isclat 18457 instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat 47834 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 18521	Extend class notation with the class of all posets.
class PosetRel
Syntax	ctsr 18522	Extend class notation with the class of all totally ordered sets.
class TosetRel
Definition	df-ps 18523	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 18524	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 18525	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 18526	A poset is a relation. (Contributed by NM, 12-May-2008.)
⊢ (𝐴 ∈ PosetRel → Rel 𝐴)
Theorem	psref2 18527	A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))
Theorem	pstr2 18528	A poset is transitive. (Contributed by FL, 3-Aug-2009.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅)
Theorem	pslem 18529	Lemma for psref 18531 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 18530	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 18531	A poset is reflexive. (Contributed by NM, 13-May-2008.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)
Theorem	psrn 18532	The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
Theorem	psasym 18533	A poset is antisymmetric. (Contributed by NM, 12-May-2008.)
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵)
Theorem	pstr 18534	A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.)
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
Theorem	cnvps 18535	The converse of a poset is a poset. In the general case (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18536 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 18536	The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)
⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel))
Theorem	psss 18537	Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.)
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
Theorem	psssdm2 18538	Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ 𝑋 = dom 𝑅    ⇒   ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴))
Theorem	psssdm 18539	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 18540	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 18541*	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 18542	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 18543	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 18544	A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.)
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
Theorem	cnvtsr 18545	The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.)
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel )
Theorem	tsrss 18546	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 18547	The domain of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.)
⊢ ℝ* = dom ≤
Theorem	lern 18548	The range of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
⊢ ℝ* = ran ≤
Theorem	lefld 18549	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 18550	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 18551	Extend class notation with the class of directed sets.
class DirRel
Syntax	ctail 18552	Extend class notation with the tail function for directed sets.
class tail
Definition	df-dir 18553	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 18554*	Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.)
⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥})))
Theorem	isdir 18555	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 18556	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 18557	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 18558	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 18559	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 18560*	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 18561	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 7407, binary operations are defined by a rule, and with df-ov 7405, 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 7405 (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 7405). The definition of magmas (Mgm, see df-mgm 18565) 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 18562	Extend class notation with group addition as a function.
class +𝑓
Syntax	cmgm 18563	Extend class notation with class of all magmas.
class Mgm
Definition	df-plusf 18564*	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 18575), while +g only has closure (mgmcl 18568). (Contributed by Mario Carneiro, 14-Aug-2015.)
⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦)))
Definition	df-mgm 18565*	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 18566*	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 18567*	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 18568	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 18569	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 18570	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 19065. (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 18571*	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 18572	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 18573	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 18574	The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ ⨣ = (+𝑓‘𝐺)    ⇒   ⊢ ⨣ Fn (𝐵 × 𝐵)
Theorem	mgmplusf 18575	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 18576*	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 18577*	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 18578*	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 18579	The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20619. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.)
⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}))
Theorem	mgmb1mgm1 18580	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 18581	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 18582	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 18583	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 18584*	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 18585) is an important property of monoids (see mndid 18669), and therefore also for groups (see grpid 18897), 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 17388. Related theorems which are already valid for magmas are provided in the following.
Theorem	mgmidmo 18585*	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 18586*	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 18587*	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 18588	The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.)
⊢ 0g Fn V
Theorem	0g0 18589	The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.)
⊢ ∅ = (0g‘∅)
Theorem	ismgmid 18590*	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 18591*	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)
Theorem	mgmlrid 18592*	The identity element of a magma, if it exists, is a left and right identity. (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 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
Theorem	ismgmid2 18593*	Show that a given element is the identity element of a magma. (Contributed by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝑈 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑈 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑈) = 𝑥)    ⇒   ⊢ (𝜑 → 𝑈 = 0 )
Theorem	lidrideqd 18594*	If there is a left and right identity element for any binary operation (group operation) +, both identity elements are equal. Generalization of statement in [Lang] p. 3: it is sufficient that "e" is a left identity element and "e`" is a right identity element instead of both being (two-sided) identity elements. (Contributed by AV, 26-Dec-2023.)
⊢ (𝜑 → 𝐿 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    ⇒   ⊢ (𝜑 → 𝐿 = 𝑅)
Theorem	lidrididd 18595*	If there is a left and right identity element for any binary operation (group operation) +, the left identity element (and therefore also the right identity element according to lidrideqd 18594) is equal to the two-sided identity element. (Contributed by AV, 26-Dec-2023.)
⊢ (𝜑 → 𝐿 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ 𝐵)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝐿 + 𝑥) = 𝑥)    &   ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑥 + 𝑅) = 𝑥)    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    ⇒   ⊢ (𝜑 → 𝐿 = 0 )
Theorem	grpidd 18596*	Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐺))    &   ⊢ (𝜑 → + = (+g‘𝐺))    &   ⊢ (𝜑 → 0 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)    ⇒   ⊢ (𝜑 → 0 = (0g‘𝐺))
Theorem	mgmidsssn0 18597*	Property of the set of identities of 𝐺. Either 𝐺 has no identities, and 𝑂 = ∅, or it has one and this identity is unique and identified by the 0g function. (Contributed by Mario Carneiro, 7-Dec-2014.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}    ⇒   ⊢ (𝐺 ∈ 𝑉 → 𝑂 ⊆ { 0 })
Theorem	grprinvlem 18598*	Lemma for grpinva 18599. (Contributed by NM, 9-Aug-2013.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → 𝑂 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑋) = 𝑋)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 = 𝑂)
Theorem	grpinva 18599*	Deduce right inverse from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → 𝑂 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → 𝑁 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝜓) → (𝑁 + 𝑋) = 𝑂)    ⇒   ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑁) = 𝑂)
Theorem	grprida 18600*	Deduce right identity from left inverse and left identity in an associative structure (such as a group). (Contributed by NM, 10-Aug-2013.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)    &   ⊢ (𝜑 → 𝑂 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 𝑂) = 𝑥)