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Type | Label | Description |
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Statement | ||
Theorem | isacs4lem 18601* | 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 18602* | 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 18603 | 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 18604 | 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 18605* | 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 18606* | 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 18607* | 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 18608 | In an algebraic closure system, the closure of a set is the union of the closures of its finite subsets. Deduction form of acsficl 18604. (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 18609* | 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 18604. 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 18610 | 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 18611* | 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 18609 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 18612* | 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 18611 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 17684, ∪ 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 18613 | 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 9384 to the map given in acsmap2d 18612. 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 18614 | 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 18613 and then applying unirnfdomd 10604 to the map given in acsmap2d 18612. 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 18615 | 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 18614 twice with acsinfd 18613. 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 18616* | In an algebraic closure system whose closure operator has the exchange property, if two independent sets have equal closure, they are equinumerous. See mreexfidimd 17694 for the finite case and acsinfdimd 18615 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 18617 | Greatest lower bounds in a Moore space are realized by intersections. (Contributed by Stefan O'Rear, 31-Jan-2015.) See mrelatglbALT 48784 for an alternate proof. |
⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐺 = (glb‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶 ∧ 𝑈 ≠ ∅) → (𝐺‘𝑈) = ∩ 𝑈) | ||
Theorem | mrelatglb0 18618 | 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 18619 | 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 48783 for an alternate proof. |
⊢ 𝐼 = (toInc‘𝐶) & ⊢ 𝐹 = (mrCls‘𝐶) & ⊢ 𝐿 = (lub‘𝐼) ⇒ ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑈 ⊆ 𝐶) → (𝐿‘𝑈) = (𝐹‘∪ 𝑈)) | ||
Theorem | mreclatBAD 18620* | A Moore space is a complete lattice under inclusion. (Contributed by Stefan O'Rear, 31-Jan-2015.) TODO (df-riota 7387 update): Reprove using isclat 18557 instead of the isclatBAD. hypothesis. See commented-out mreclat above. See mreclat 48785 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) | ||
See commented-out notes for lattices as relations. | ||
Syntax | cps 18621 | Extend class notation with the class of all posets. |
class PosetRel | ||
Syntax | ctsr 18622 | Extend class notation with the class of all totally ordered sets. |
class TosetRel | ||
Definition | df-ps 18623 | 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 18624 | 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 18625 | 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 18626 | A poset is a relation. (Contributed by NM, 12-May-2008.) |
⊢ (𝐴 ∈ PosetRel → Rel 𝐴) | ||
Theorem | psref2 18627 | A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | pstr2 18628 | A poset is transitive. (Contributed by FL, 3-Aug-2009.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | ||
Theorem | pslem 18629 | Lemma for psref 18631 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 18630 | 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 18631 | A poset is reflexive. (Contributed by NM, 13-May-2008.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) | ||
Theorem | psrn 18632 | The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) | ||
Theorem | psasym 18633 | A poset is antisymmetric. (Contributed by NM, 12-May-2008.) |
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵) | ||
Theorem | pstr 18634 | A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | ||
Theorem | cnvps 18635 | The converse of a poset is a poset. In the general case (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18636 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 18636 | The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.) |
⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel)) | ||
Theorem | psss 18637 | Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel) | ||
Theorem | psssdm2 18638 | Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)) | ||
Theorem | psssdm 18639 | 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 18640 | 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 18641* | 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 18642 | 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 18643 | 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 18644 | A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) | ||
Theorem | cnvtsr 18645 | The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) | ||
Theorem | tsrss 18646 | 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 18647 | The domain of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.) |
⊢ ℝ* = dom ≤ | ||
Theorem | lern 18648 | The range of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
⊢ ℝ* = ran ≤ | ||
Theorem | lefld 18649 | 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 18650 | 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 | ||
Syntax | cdir 18651 | Extend class notation with the class of directed sets. |
class DirRel | ||
Syntax | ctail 18652 | Extend class notation with the tail function for directed sets. |
class tail | ||
Definition | df-dir 18653 | 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 18654* | Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.) |
⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥}))) | ||
Theorem | isdir 18655 | 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 18656 | 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 18657 | 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 18658 | 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 18659 | 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 18660* | 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 18661 | 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) | ||
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 7435, binary operations are defined by a rule, and with df-ov 7433, 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 7433 (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 7433). The definition of magmas (Mgm, see df-mgm 18665) 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 18662 | Extend class notation with group addition as a function. |
class +𝑓 | ||
Syntax | cmgm 18663 | Extend class notation with class of all magmas. |
class Mgm | ||
Definition | df-plusf 18664* | 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 18675), while +g only has closure (mgmcl 18668). (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) | ||
Definition | df-mgm 18665* | 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 18666* | 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 18667* | 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 18668 | 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 18669 | 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 18670 | 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 19176. (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 18671* | 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 18672 | 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 18673 | 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 18674 | The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⨣ = (+𝑓‘𝐺) ⇒ ⊢ ⨣ Fn (𝐵 × 𝐵) | ||
Theorem | mgmplusf 18675 | 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 18676* | 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 18677* | 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 18678* | 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 18679 | The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20793. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.) |
⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {〈〈𝑍, 𝑍〉, 𝑍〉})) | ||
Theorem | mgmb1mgm1 18680 | 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 18681 | 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 18682 | 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 18683 | 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 18684* | 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) | ||
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 18685) is an important property of monoids (see mndid 18769), and therefore also for groups (see grpid 19005), 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 17487. Related theorems which are already valid for magmas are provided in the following. | ||
Theorem | mgmidmo 18685* | 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 18686* | 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 18687* | 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 18688 | The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 0g Fn V | ||
Theorem | 0g0 18689 | The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.) |
⊢ ∅ = (0g‘∅) | ||
Theorem | ismgmid 18690* | 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 18691* | 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 18692* | 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 18693* | 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 18694* | 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 18695* | 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 18694) 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 18696* | 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 18697* | 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 | grpinvalem 18698* | Lemma for grpinva 18699. (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 18699* | 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 18700* | 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) |
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