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Type | Label | Description |
---|---|---|
Statement | ||
Syntax | cdlat 17801 | The class of distributive lattices. |
class DLat | ||
Definition | df-dlat 17802* | A distributive lattice is a lattice in which meets distribute over joins, or equivalently (latdisd 17800) joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ DLat = {𝑘 ∈ Lat ∣ [(Base‘𝑘) / 𝑏][(join‘𝑘) / 𝑗][(meet‘𝑘) / 𝑚]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∀𝑧 ∈ 𝑏 (𝑥𝑚(𝑦𝑗𝑧)) = ((𝑥𝑚𝑦)𝑗(𝑥𝑚𝑧))} | ||
Theorem | isdlat 17803* | Property of being a distributive lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ (𝐾 ∈ DLat ↔ (𝐾 ∈ Lat ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 (𝑥 ∧ (𝑦 ∨ 𝑧)) = ((𝑥 ∧ 𝑦) ∨ (𝑥 ∧ 𝑧)))) | ||
Theorem | dlatmjdi 17804 | In a distributive lattice, meets distribute over joins. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∧ (𝑌 ∨ 𝑍)) = ((𝑋 ∧ 𝑌) ∨ (𝑋 ∧ 𝑍))) | ||
Theorem | dlatl 17805 | A distributive lattice is a lattice. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ (𝐾 ∈ DLat → 𝐾 ∈ Lat) | ||
Theorem | odudlatb 17806 | The dual of a distributive lattice is a distributive lattice and conversely. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐷 = (ODual‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑉 → (𝐾 ∈ DLat ↔ 𝐷 ∈ DLat)) | ||
Theorem | dlatjmdi 17807 | In a distributive lattice, joins distribute over meets. (Contributed by Stefan O'Rear, 30-Jan-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) ⇒ ⊢ ((𝐾 ∈ DLat ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ∨ (𝑌 ∧ 𝑍)) = ((𝑋 ∨ 𝑌) ∧ (𝑋 ∨ 𝑍))) | ||
Syntax | cps 17808 | Extend class notation with the class of all posets. |
class PosetRel | ||
Syntax | ctsr 17809 | Extend class notation with the class of all totally ordered sets. |
class TosetRel | ||
Definition | df-ps 17810 | 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 17811 | 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 17812 | 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 17813 | A poset is a relation. (Contributed by NM, 12-May-2008.) |
⊢ (𝐴 ∈ PosetRel → Rel 𝐴) | ||
Theorem | psref2 17814 | A poset is antisymmetric and reflexive. (Contributed by FL, 3-Aug-2009.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | ||
Theorem | pstr2 17815 | A poset is transitive. (Contributed by FL, 3-Aug-2009.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | ||
Theorem | pslem 17816 | Lemma for psref 17818 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 17817 | 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 17818 | A poset is reflexive. (Contributed by NM, 13-May-2008.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) | ||
Theorem | psrn 17819 | The range of a poset equals it domain. (Contributed by NM, 7-Jul-2008.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅) | ||
Theorem | psasym 17820 | A poset is antisymmetric. (Contributed by NM, 12-May-2008.) |
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐴) → 𝐴 = 𝐵) | ||
Theorem | pstr 17821 | A poset is transitive. (Contributed by NM, 12-May-2008.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ ((𝑅 ∈ PosetRel ∧ 𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | ||
Theorem | cnvps 17822 | The converse of a poset is a poset. In the general case (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 17823 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 17823 | The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.) |
⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel)) | ||
Theorem | psss 17824 | Any subset of a partially ordered set is partially ordered. (Contributed by FL, 24-Jan-2010.) |
⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel) | ||
Theorem | psssdm2 17825 | Field of a subposet. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ 𝑋 = dom 𝑅 ⇒ ⊢ (𝑅 ∈ PosetRel → dom (𝑅 ∩ (𝐴 × 𝐴)) = (𝑋 ∩ 𝐴)) | ||
Theorem | psssdm 17826 | 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 17827 | 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 17828* | 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 17829 | 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 17830 | 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 17831 | A toset is a poset. (Contributed by Mario Carneiro, 9-Sep-2015.) |
⊢ (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel) | ||
Theorem | cnvtsr 17832 | The converse of a toset is a toset. (Contributed by Mario Carneiro, 3-Sep-2015.) |
⊢ (𝑅 ∈ TosetRel → ◡𝑅 ∈ TosetRel ) | ||
Theorem | tsrss 17833 | 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 17834 | The domain of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 4-May-2015.) |
⊢ ℝ* = dom ≤ | ||
Theorem | lern 17835 | The range of ≤ is ℝ*. (Contributed by FL, 2-Aug-2009.) (Revised by Mario Carneiro, 3-Sep-2015.) |
⊢ ℝ* = ran ≤ | ||
Theorem | lefld 17836 | 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 17837 | 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 17838 | Extend class notation with the class of directed sets. |
class DirRel | ||
Syntax | ctail 17839 | Extend class notation with the tail function for directed sets. |
class tail | ||
Definition | df-dir 17840 | 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 17841* | Define the tail function for directed sets. (Contributed by Jeff Hankins, 25-Nov-2009.) |
⊢ tail = (𝑟 ∈ DirRel ↦ (𝑥 ∈ ∪ ∪ 𝑟 ↦ (𝑟 “ {𝑥}))) | ||
Theorem | isdir 17842 | 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 17843 | 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 17844 | 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 17845 | 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 17846 | 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 17847* | 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 17848 | 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 7161, binary operations are defined by a rule, and with df-ov 7159, 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 7159 (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 7159). The definition of magmas (Mgm, see df-mgm 17852) 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 17849 | Extend class notation with group addition as a function. |
class +𝑓 | ||
Syntax | cmgm 17850 | Extend class notation with class of all magmas. |
class Mgm | ||
Definition | df-plusf 17851* | 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 17862), while +g only has closure (mgmcl 17855). (Contributed by Mario Carneiro, 14-Aug-2015.) |
⊢ +𝑓 = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(+g‘𝑔)𝑦))) | ||
Definition | df-mgm 17852* | 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 17853* | 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 17854* | 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 17855 | 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 17856 | 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 17857 | 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 18299. (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 17858* | 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 17859 | 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 17860 | 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 17861 | The group addition operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ ⨣ = (+𝑓‘𝐺) ⇒ ⊢ ⨣ Fn (𝐵 × 𝐵) | ||
Theorem | mgmplusf 17862 | 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 | issstrmgm 17863* | 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 17864 | The internal operation for a set is the trivial operation iff the set is a singleton. Formerly part of proof of ring1zr 20048. (Contributed by FL, 13-Feb-2010.) (Revised by AV, 23-Jan-2020.) |
⊢ (( ⚬ :(𝐵 × 𝐵)⟶𝐵 ∧ 𝑍 ∈ 𝐵) → (𝐵 = {𝑍} ↔ ⚬ = {〈〈𝑍, 𝑍〉, 𝑍〉})) | ||
Theorem | mgmb1mgm1 17865 | 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 17866 | 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 17867 | 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 17868 | 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 17869* | 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 17870) is an important property of monoids (see mndid 17921), and therefore also for groups (see grpid 18139), 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 16715. Related theorems which are already valid for magmas are provided in the following. | ||
Theorem | mgmidmo 17870* | 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 17871* | 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 17872* | 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 17873 | The group zero extractor is a function. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 0g Fn V | ||
Theorem | 0g0 17874 | The identity element function evaluates to the empty set on an empty structure. (Contributed by Stefan O'Rear, 2-Oct-2015.) |
⊢ ∅ = (0g‘∅) | ||
Theorem | ismgmid 17875* | 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 17876* | 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 17877* | 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 17878* | 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 17879* | 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 17880* | 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 17879) 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 17881* | 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 17882* | 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 17883* | Lemma for grprinvd 17884. (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 | grprinvd 17884* | 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 | grpridd 17885* | 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) | ||
The symbol Σg is mostly used in the context of abelian groups. Therefore, it is usually called "group sum". It can be defined, however, in arbitrary magmas (then it should be called "iterated sum"). If the magma is not required to be commutative or associative, then the order of the summands and the order in which summations are done become important. If the magma is not unital, then one cannot define a meaningful empty sum. See Remark 2. in the comment for df-gsum 16716. | ||
Theorem | gsumvalx 17886* | Expand out the substitutions in df-gsum 16716. (Contributed by Mario Carneiro, 18-Sep-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)} & ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ 𝑂))) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ 𝑋) & ⊢ (𝜑 → dom 𝐹 = 𝐴) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))))) | ||
Theorem | gsumval 17887* | Expand out the substitutions in df-gsum 16716. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝑂 = {𝑠 ∈ 𝐵 ∣ ∀𝑡 ∈ 𝐵 ((𝑠 + 𝑡) = 𝑡 ∧ (𝑡 + 𝑠) = 𝑡)} & ⊢ (𝜑 → 𝑊 = (◡𝐹 “ (V ∖ 𝑂))) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, 0 , if(𝐴 ∈ ran ..., (℩𝑥∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)(𝐴 = (𝑚...𝑛) ∧ 𝑥 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑥∃𝑓(𝑓:(1...(♯‘𝑊))–1-1-onto→𝑊 ∧ 𝑥 = (seq1( + , (𝐹 ∘ 𝑓))‘(♯‘𝑊))))))) | ||
Theorem | gsumpropd 17888 | The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 17936 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ (𝜑 → (+g‘𝐺) = (+g‘𝐻)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumpropd2lem 17889* | Lemma for gsumpropd2 17890. (Contributed by Thierry Arnoux, 28-Jun-2017.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) & ⊢ 𝐴 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐺) ∣ ∀𝑡 ∈ (Base‘𝐺)((𝑠(+g‘𝐺)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐺)𝑠) = 𝑡)})) & ⊢ 𝐵 = (◡𝐹 “ (V ∖ {𝑠 ∈ (Base‘𝐻) ∣ ∀𝑡 ∈ (Base‘𝐻)((𝑠(+g‘𝐻)𝑡) = 𝑡 ∧ (𝑡(+g‘𝐻)𝑠) = 𝑡)})) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumpropd2 17890* | A stronger version of gsumpropd 17888, working for magma, where only the closure of the addition operation on a common base is required, see gsummgmpropd 17891. (Contributed by Thierry Arnoux, 28-Jun-2017.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) ∈ (Base‘𝐺)) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsummgmpropd 17891* | A stronger version of gsumpropd 17888 if at least one of the involved structures is a magma, see gsumpropd2 17890. (Contributed by AV, 31-Jan-2020.) |
⊢ (𝜑 → 𝐹 ∈ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑊) & ⊢ (𝜑 → 𝐻 ∈ 𝑋) & ⊢ (𝜑 → (Base‘𝐺) = (Base‘𝐻)) & ⊢ (𝜑 → 𝐺 ∈ Mgm) & ⊢ ((𝜑 ∧ (𝑠 ∈ (Base‘𝐺) ∧ 𝑡 ∈ (Base‘𝐺))) → (𝑠(+g‘𝐺)𝑡) = (𝑠(+g‘𝐻)𝑡)) & ⊢ (𝜑 → Fun 𝐹) & ⊢ (𝜑 → ran 𝐹 ⊆ (Base‘𝐺)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumress 17892* | The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither 𝐺 nor 𝐻 need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐻 = (𝐺 ↾s 𝑆) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑋) & ⊢ (𝜑 → 𝑆 ⊆ 𝐵) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑆) & ⊢ (𝜑 → 0 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (𝐻 Σg 𝐹)) | ||
Theorem | gsumval1 17893* | Value of the group sum operation when every element being summed is an identity of 𝐺. (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)} & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐴 ∈ 𝑊) & ⊢ (𝜑 → 𝐹:𝐴⟶𝑂) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = 0 ) | ||
Theorem | gsum0 17894 | Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 Σg ∅) = 0 | ||
Theorem | gsumval2a 17895* | Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) & ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} & ⊢ (𝜑 → ¬ ran 𝐹 ⊆ 𝑂) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | gsumval2 17896 | Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁)) | ||
Theorem | gsumsplit1r 17897 | Splitting off the rightmost summand of a group sum. This corresponds to the (inductive) definition of a (finite) product in [Lang] p. 4, first formula. (Contributed by AV, 26-Dec-2023.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) & ⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐺 Σg (𝐹 ↾ (𝑀...𝑁))) + (𝐹‘(𝑁 + 1)))) | ||
Theorem | gsumprval 17898 | Value of the group sum operation over a pair of sequential integers. (Contributed by AV, 14-Dec-2018.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝑁 = (𝑀 + 1)) & ⊢ (𝜑 → 𝐹:{𝑀, 𝑁}⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘𝑀) + (𝐹‘𝑁))) | ||
Theorem | gsumpr12val 17899 | Value of the group sum operation over the pair {1, 2}. (Contributed by AV, 14-Dec-2018.) |
⊢ 𝐵 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ 𝑉) & ⊢ (𝜑 → 𝐹:{1, 2}⟶𝐵) ⇒ ⊢ (𝜑 → (𝐺 Σg 𝐹) = ((𝐹‘1) + (𝐹‘2))) | ||
A semigroup (Smgrp, see df-sgrp 17901) is a set together with an associative binary operation (see Wikipedia, Semigroup, 8-Jan-2020, https://en.wikipedia.org/wiki/Semigroup 17901). In other words, a semigroup is an associative magma. The notion of semigroup is a generalization of that of group where the existence of an identity or inverses is not required. | ||
Syntax | csgrp 17900 | Extend class notation with class of all semigroups. |
class Smgrp |
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