Home Metamath Proof ExplorerTheorem List (p. 422 of 429) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-27903) Hilbert Space Explorer (27904-29428) Users' Mathboxes (29429-42879)

Theorem List for Metamath Proof Explorer - 42101-42200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremismgmd 42101* Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.)
(𝜑𝐵 = (Base‘𝐺))    &   (𝜑𝐺𝑉)    &   (𝜑+ = (+g𝐺))    &   ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)       (𝜑𝐺 ∈ Mgm)

20.35.11.3  Magma homomorphisms and submagmas

Syntaxcmgmhm 42102 Hom-set generator class for magmas.
class MgmHom

Syntaxcsubmgm 42103 Class function taking a magma to its lattice of submagmas.
class SubMgm

Definitiondf-mgmhm 42104* A magma homomorphism is a function on the base sets which preserves the binary operation. (Contributed by AV, 24-Feb-2020.)
MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦))})

Definitiondf-submgm 42105* A submagma is a subset of a magma which is closed under the operation. Such subsets are themselves magmas. (Contributed by AV, 24-Feb-2020.)
SubMgm = (𝑠 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑠) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑠)𝑦) ∈ 𝑡})

Theoremmgmhmrcl 42106 Reverse closure of a magma homomorphism. (Contributed by AV, 24-Feb-2020.)
(𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))

Theoremsubmgmrcl 42107 Reverse closure for submagmas. (Contributed by AV, 24-Feb-2020.)
(𝑆 ∈ (SubMgm‘𝑀) → 𝑀 ∈ Mgm)

Theoremismgmhm 42108* Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)       (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))

Theoremmgmhmf 42109 A magma homomorphism is a function. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       (𝐹 ∈ (𝑆 MgmHom 𝑇) → 𝐹:𝐵𝐶)

Theoremmgmhmpropd 42110* Magma homomorphism depends only on the operation of structures. (Contributed by AV, 25-Feb-2020.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   (𝜑𝐵 ≠ ∅)    &   (𝜑𝐶 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))       (𝜑 → (𝐽 MgmHom 𝐾) = (𝐿 MgmHom 𝑀))

Theoremmgmhmlin 42111 A magma homomorphism preserves the binary operation. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑆)    &    + = (+g𝑆)    &    = (+g𝑇)       ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘(𝑋 + 𝑌)) = ((𝐹𝑋) (𝐹𝑌)))

Theoremmgmhmf1o 42112 A magma homomorphism is bijective iff its converse is also a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 MgmHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 MgmHom 𝑅)))

Theoremidmgmhm 42113 The identity homomorphism on a magma. (Contributed by AV, 27-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mgm → ( I ↾ 𝐵) ∈ (𝑀 MgmHom 𝑀))

Theoremissubmgm 42114* Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Theoremissubmgm2 42115 Submagmas are subsets that are also magmas. (Contributed by AV, 25-Feb-2020.)
𝐵 = (Base‘𝑀)    &   𝐻 = (𝑀s 𝑆)       (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵𝐻 ∈ Mgm)))

Theoremrabsubmgmd 42116* Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &   (𝜑𝑀 ∈ Mgm)    &   ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)    &   (𝑧 = 𝑥 → (𝜓𝜃))    &   (𝑧 = 𝑦 → (𝜓𝜏))    &   (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))       (𝜑 → {𝑧𝐵𝜓} ∈ (SubMgm‘𝑀))

Theoremsubmgmss 42117 Submagmas are subsets of the base set. (Contributed by AV, 26-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑆 ∈ (SubMgm‘𝑀) → 𝑆𝐵)

Theoremsubmgmid 42118 Every magma is trivially a submagma of itself. (Contributed by AV, 26-Feb-2020.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ Mgm → 𝐵 ∈ (SubMgm‘𝑀))

Theoremsubmgmcl 42119 Submagmas are closed under the monoid operation. (Contributed by AV, 26-Feb-2020.)
+ = (+g𝑀)       ((𝑆 ∈ (SubMgm‘𝑀) ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Theoremsubmgmmgm 42120 Submagmas are themselves magmas under the given operation. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMgm‘𝑀) → 𝐻 ∈ Mgm)

Theoremsubmgmbas 42121 The base set of a submagma. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝑀s 𝑆)       (𝑆 ∈ (SubMgm‘𝑀) → 𝑆 = (Base‘𝐻))

Theoremsubsubmgm 42122 A submagma of a submagma is a submagma. (Contributed by AV, 26-Feb-2020.)
𝐻 = (𝐺s 𝑆)       (𝑆 ∈ (SubMgm‘𝐺) → (𝐴 ∈ (SubMgm‘𝐻) ↔ (𝐴 ∈ (SubMgm‘𝐺) ∧ 𝐴𝑆)))

Theoremresmgmhm 42123 Restriction of a magma homomorphism to a submagma is a homomorphism. (Contributed by AV, 26-Feb-2020.)
𝑈 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝑋 ∈ (SubMgm‘𝑆)) → (𝐹𝑋) ∈ (𝑈 MgmHom 𝑇))

Theoremresmgmhm2 42124 One direction of resmgmhm2b 42125. (Contributed by AV, 26-Feb-2020.)
𝑈 = (𝑇s 𝑋)       ((𝐹 ∈ (𝑆 MgmHom 𝑈) ∧ 𝑋 ∈ (SubMgm‘𝑇)) → 𝐹 ∈ (𝑆 MgmHom 𝑇))

Theoremresmgmhm2b 42125 Restriction of the codomain of a homomorphism. (Contributed by AV, 26-Feb-2020.)
𝑈 = (𝑇s 𝑋)       ((𝑋 ∈ (SubMgm‘𝑇) ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ (𝑆 MgmHom 𝑈)))

Theoremmgmhmco 42126 The composition of magma homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑇 MgmHom 𝑈) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 MgmHom 𝑈))

Theoremmgmhmima 42127 The homomorphic image of a submagma is a submagma. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑀 MgmHom 𝑁) ∧ 𝑋 ∈ (SubMgm‘𝑀)) → (𝐹𝑋) ∈ (SubMgm‘𝑁))

Theoremmgmhmeql 42128 The equalizer of two magma homomorphisms is a submagma. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ 𝐺 ∈ (𝑆 MgmHom 𝑇)) → dom (𝐹𝐺) ∈ (SubMgm‘𝑆))

Theoremsubmgmacs 42129 Submagmas are an algebraic closure system. (Contributed by AV, 27-Feb-2020.)
𝐵 = (Base‘𝐺)       (𝐺 ∈ Mgm → (SubMgm‘𝐺) ∈ (ACS‘𝐵))

Theoremismhm0 42130 Property of a monoid homomorphism, expressed by a magma homomorphism. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &    + = (+g𝑆)    &    = (+g𝑇)    &    0 = (0g𝑆)    &   𝑌 = (0g𝑇)       (𝐹 ∈ (𝑆 MndHom 𝑇) ↔ ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) ∧ (𝐹 ∈ (𝑆 MgmHom 𝑇) ∧ (𝐹0 ) = 𝑌)))

Theoremmhmismgmhm 42131 Each monoid homomorphism is a magma homomorphism. (Contributed by AV, 29-Feb-2020.)
(𝐹 ∈ (𝑅 MndHom 𝑆) → 𝐹 ∈ (𝑅 MgmHom 𝑆))

20.35.11.4  Examples and counterexamples for magmas, semigroups and monoids (extension)

Theoremopmpt2ismgm 42132* A structure with a group addition operation in maps-to notation is a magma if the operation value is contained in the base set. (Contributed by AV, 16-Feb-2020.)
𝐵 = (Base‘𝑀)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)    &   (𝜑𝐵 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → 𝐶𝐵)       (𝜑𝑀 ∈ Mgm)

Theoremcopissgrp 42133* A structure with a constant group addition operation is a semigroup if the constant is contained in the base set. (Contributed by AV, 16-Feb-2020.)
𝐵 = (Base‘𝑀)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)    &   (𝜑𝐵 ≠ ∅)    &   (𝜑𝐶𝐵)       (𝜑𝑀 ∈ SGrp)

Theoremcopisnmnd 42134* A structure with a constant group addition operation and at least two elements is not a monoid. (Contributed by AV, 16-Feb-2020.)
𝐵 = (Base‘𝑀)    &   (+g𝑀) = (𝑥𝐵, 𝑦𝐵𝐶)    &   (𝜑𝐶𝐵)    &   (𝜑 → 1 < (#‘𝐵))       (𝜑𝑀 ∉ Mnd)

Theorem0nodd 42135* 0 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}       0 ∉ 𝑂

Theorem1odd 42136* 1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}       1 ∈ 𝑂

Theorem2nodd 42137* 2 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}       2 ∉ 𝑂

Theoremoddibas 42138* Lemma 1 for oddinmgm 42140: The base set of M is the set of all odd integers. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   𝑀 = (ℂflds 𝑂)       𝑂 = (Base‘𝑀)

Theoremoddiadd 42139* Lemma 2 for oddinmgm 42140: The group addition operation of M is the addition of complex numbers. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   𝑀 = (ℂflds 𝑂)        + = (+g𝑀)

Theoremoddinmgm 42140* The structure of all odd integers together with the addition of complex numbers is not a magma. Remark: the structure of the complementary subset of the set of integers, the even integers, is a magma, actually an abelian group, see 2zrngaabl 42269, and even a non-unital ring, see 2zrng 42260. (Contributed by AV, 3-Feb-2020.)
𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   𝑀 = (ℂflds 𝑂)       𝑀 ∉ Mgm

Theoremnnsgrpmgm 42141 The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
𝑀 = (ℂflds ℕ)       𝑀 ∈ Mgm

Theoremnnsgrp 42142 The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
𝑀 = (ℂflds ℕ)       𝑀 ∈ SGrp

Theoremnnsgrpnmnd 42143 The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.)
𝑀 = (ℂflds ℕ)       𝑀 ∉ Mnd

20.35.12  Magmas and internal binary operations (alternate approach)

With df-mpt2 6695, binary operations are defined by a rule, and with df-ov 6693, 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 (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 × 𝑆𝑆. 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, we call binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 internal binary operations, see df-intop 42160. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 42161. Therefore, a "binary operation on a set 𝑆" according to Wikipedia is a "closed internal binary operation" in our 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 ).

Taking a step back, we define "laws" applicable for "binary operations" (which even need not to be functions), according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7. These laws are used, on the one hand, to specialize internal binary operations (see df-clintop 42161 and df-assintop 42162), and on the other hand to define the common algebraic structures like magmas, groups, rings, etc. Internal binary operations, which obey these laws, are defined afterwards. Notice that in [BourbakiAlg1] p. 1, p. 4 and p. 7, these operations are called "laws" by themselves.

In the following, an alternate definition df-cllaw 42147 for an internal binary operation is provided, which does not require function-ness, but only closure. Therefore, this definition could be used as binary operation (Slot 2) defined for a magma as extensible structure, see mgmplusgiopALT 42155, or for an alternate definition df-mgm2 42180 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups).

20.35.12.1  Laws for internal binary operations

In this subsection, the "laws" applicable for "binary operations" according to the definition in [Hall] p. 1 and [BourbakiAlg1] p. 1, p. 4 and p. 7 are defined. These laws are called "internal laws" in [BourbakiAlg1] p. xxi.

Syntaxccllaw 42144 Extend class notation for the closure law.
class clLaw

Syntaxcasslaw 42145 Extend class notation for the associative law.
class assLaw

Syntaxccomlaw 42146 Extend class notation for the commutative law.
class comLaw

Definitiondf-cllaw 42147* The closure law for binary operations, see definitions of laws A0. and M0. in section 1.1 of [Hall] p. 1, or definition 1 in [BourbakiAlg1] p. 1: the value of a binary operation applied to two operands of a given sets is an element of this set. By this definition, the closure law is expressed as binary relation: a binary operation is related to a set by clLaw if the closure law holds for this binary operation regarding this set. Note that the binary operation needs not to be a function. (Contributed by AV, 7-Jan-2020.)
clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}

Definitiondf-comlaw 42148* The commutative law for binary operations, see definitions of laws A2. and M2. in section 1.1 of [Hall] p. 1, or definition 8 in [BourbakiAlg1] p. 7: the value of a binary operation applied to two operands equals the value of a binary operation applied to the two operands in reversed order. By this definition, the commutative law is expressed as binary relation: a binary operation is related to a set by comLaw if the commutative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by AV, 7-Jan-2020.)
comLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}

Definitiondf-asslaw 42149* The associative law for binary operations, see definitions of laws A1. and M1. in section 1.1 of [Hall] p. 1, or definition 5 in [BourbakiAlg1] p. 4: the value of a binary operation applied the value of the binary operation applied to two operands and a third operand equals the value of the binary operation applied to the first operand and the value of the binary operation applied to the second and third operand. By this definition, the associative law is expressed as binary relation: a binary operation is related to a set by assLaw if the associative law holds for this binary operation regarding this set. Note that the binary operation needs neither to be closed nor to be a function. (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
assLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚𝑧𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}

Theoremiscllaw 42150* The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)
(( 𝑉𝑀𝑊) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))

Theoremiscomlaw 42151* The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.)
(( 𝑉𝑀𝑊) → ( comLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))

Theoremclcllaw 42152 Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)
(( clLaw 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)

Theoremisasslaw 42153* The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
(( 𝑉𝑀𝑊) → ( assLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))

Theoremasslawass 42154* Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.)
( assLaw 𝑀 → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))

TheoremmgmplusgiopALT 42155 Slot 2 (group operation) of a magma as extensible structure is a closed operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝑀 ∈ Mgm → (+g𝑀) clLaw (Base‘𝑀))

TheoremsgrpplusgaopALT 42156 Slot 2 (group operation) of a semigroup as extensible structure is an associative operation on the base set. (Contributed by AV, 13-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝐺 ∈ SGrp → (+g𝐺) assLaw (Base‘𝐺))

20.35.12.2  Internal binary operations

In this subsection, "internal binary operations" obeying different laws are defined.

Syntaxcintop 42157 Extend class notation with class of internal (binary) operations for a set.
class intOp

Syntaxcclintop 42158 Extend class notation with class of closed operations for a set.
class clIntOp

Syntaxcassintop 42159 Extend class notation with class of associative operations for a set.
class assIntOp

Definitiondf-intop 42160* Function mapping a set to the class of all internal (binary) operations for this set. (Contributed by AV, 20-Jan-2020.)
intOp = (𝑚 ∈ V, 𝑛 ∈ V ↦ (𝑛𝑚 (𝑚 × 𝑚)))

Definitiondf-clintop 42161 Function mapping a set to the class of all closed (internal binary) operations for this set, see definition in section 1.2 of [Hall] p. 2, definition in section I.1 of [Bruck] p. 1, or definition 1 in [BourbakiAlg1] p. 1, where it is called "a law of composition". (Contributed by AV, 20-Jan-2020.)
clIntOp = (𝑚 ∈ V ↦ (𝑚 intOp 𝑚))

Definitiondf-assintop 42162* Function mapping a set to the class of all associative (closed internal binary) operations for this set, see definition 5 in [BourbakiAlg1] p. 4, where it is called "an associative law of composition". (Contributed by AV, 20-Jan-2020.)
assIntOp = (𝑚 ∈ V ↦ {𝑜 ∈ ( clIntOp ‘𝑚) ∣ 𝑜 assLaw 𝑚})

Theoremintopval 42163 The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁𝑚 (𝑀 × 𝑀)))

Theoremintop 42164 An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ (𝑀 intOp 𝑁) → :(𝑀 × 𝑀)⟶𝑁)

Theoremclintopval 42165 The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
(𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))

Theoremassintopval 42166* The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
(𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})

Theoremassintopmap 42167* The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)
(𝑀𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀𝑚 (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})

Theoremisclintop 42168 The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
(𝑀𝑉 → ( ∈ ( clIntOp ‘𝑀) ↔ :(𝑀 × 𝑀)⟶𝑀))

Theoremclintop 42169 A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)

Theoremassintop 42170 An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))

Theoremisassintop 42171* The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
(𝑀𝑉 → ( ∈ ( assIntOp ‘𝑀) ↔ ( :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))))

Theoremclintopcllaw 42172 The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)

Theoremassintopcllaw 42173 The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
( ∈ ( assIntOp ‘𝑀) → clLaw 𝑀)

Theoremassintopasslaw 42174 The associative low holds for a associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
( ∈ ( assIntOp ‘𝑀) → assLaw 𝑀)

Theoremassintopass 42175* An associative (closed internal binary) operation for a set is associative. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
( ∈ ( assIntOp ‘𝑀) → ∀𝑥𝑀𝑦𝑀𝑧𝑀 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧)))

20.35.12.3  Alternative definitions for Magmas and Semigroups

Syntaxcmgm2 42176 Extend class notation with class of all magmas.
class MgmALT

Syntaxccmgm2 42177 Extend class notation with class of all commutative magmas.
class CMgmALT

Syntaxcsgrp2 42178 Extend class notation with class of all semigroups.
class SGrpALT

Syntaxccsgrp2 42179 Extend class notation with class of all commutative semigroups.
class CSGrpALT

Definitiondf-mgm2 42180 A magma is a set equipped with a closed operation. Definition 1 of [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 AV, 6-Jan-2020.)
MgmALT = {𝑚 ∣ (+g𝑚) clLaw (Base‘𝑚)}

Definitiondf-cmgm2 42181 A commutative magma is a magma with a commutative operation. Definition 8 of [BourbakiAlg1] p. 7. (Contributed by AV, 20-Jan-2020.)
CMgmALT = {𝑚 ∈ MgmALT ∣ (+g𝑚) comLaw (Base‘𝑚)}

Definitiondf-sgrp2 42182 A semigroup is a magma with an associative operation. Definition in section II.1 of [Bruck] p. 23, or of an "associative magma" in definition 5 of [BourbakiAlg1] p. 4, or of a semi-group in section 1.3 of [Hall] p. 7. (Contributed by AV, 6-Jan-2020.)
SGrpALT = {𝑔 ∈ MgmALT ∣ (+g𝑔) assLaw (Base‘𝑔)}

Definitiondf-csgrp2 42183 A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g𝑔) comLaw (Base‘𝑔)}

TheoremismgmALT 42184 The predicate "is a magma." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀𝑉 → (𝑀 ∈ MgmALT ↔ clLaw 𝐵))

TheoremiscmgmALT 42185 The predicate "is a commutative magma." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ comLaw 𝐵))

TheoremissgrpALT 42186 The predicate "is a semigroup." (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ assLaw 𝐵))

TheoremiscsgrpALT 42187 The predicate "is a commutative semigroup." (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐵 = (Base‘𝑀)    &    = (+g𝑀)       (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ comLaw 𝐵))

Theoremmgm2mgm 42188 Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
(𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)

Theoremsgrp2sgrp 42189 Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
(𝑀 ∈ SGrpALT ↔ 𝑀 ∈ SGrp)

20.35.13  Categories (extension)

20.35.13.1  Subcategories (extension)

Theoremidfusubc0 42190* The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
𝑆 = (𝐶cat 𝐽)    &   𝐼 = (idfunc𝑆)    &   𝐵 = (Base‘𝑆)       (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = ⟨( I ↾ 𝐵), (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥(Hom ‘𝑆)𝑦)))⟩)

Theoremidfusubc 42191* The identity functor for a subcategory is an "inclusion functor" from the subcategory into its supercategory. (Contributed by AV, 29-Mar-2020.)
𝑆 = (𝐶cat 𝐽)    &   𝐼 = (idfunc𝑆)    &   𝐵 = (Base‘𝑆)       (𝐽 ∈ (Subcat‘𝐶) → 𝐼 = ⟨( I ↾ 𝐵), (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥𝐽𝑦)))⟩)

Theoreminclfusubc 42192* The "inclusion functor" from a subcategory of a category into the category itself. (Contributed by AV, 30-Mar-2020.)
(𝜑𝐽 ∈ (Subcat‘𝐶))    &   𝑆 = (𝐶cat 𝐽)    &   𝐵 = (Base‘𝑆)    &   (𝜑𝐹 = ( I ↾ 𝐵))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥𝐽𝑦))))       (𝜑𝐹(𝑆 Func 𝐶)𝐺)

20.35.14  Rings (extension)

20.35.14.1  Nonzero rings (extension)

Theoremlmod0rng 42193 If the scalar ring of a module is the zero ring, the module is the zero module, i.e. the base set of the module is the singleton consisting of the identity element only. (Contributed by AV, 17-Apr-2019.)
((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing) → (Base‘𝑀) = {(0g𝑀)})

Theoremnzrneg1ne0 42194 The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.)
(𝑅 ∈ NzRing → ((invg𝑅)‘(1r𝑅)) ≠ (0g𝑅))

Theorem0ringdif 42195 A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 }))

Theorem0ringbas 42196 The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ (Ring ∖ NzRing) → 𝐵 = { 0 })

Theorem0ring1eq0 42197 In a zero ring, a ring which is not a nonzero ring, the unit equals the zero element. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ (Ring ∖ NzRing) → 1 = 0 )

Theoremnrhmzr 42198 There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.)
((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅)

20.35.14.2  Non-unital rings ("rngs")

According to Wikipedia, "... in abstract algebra, a rng (or pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a [unital] ring, without assuming the existence of a multiplicative identity. The term "rng" (pronounced rung) is meant to suggest that it is a "ring" without "i", i.e. without the requirement for an "identity element"." (see https://en.wikipedia.org/wiki/Rng_(algebra), 6-Jan-2020).

Syntaxcrng 42199 Extend class notation with class of all non-unital rings.
class Rng

Definitiondf-rng0 42200* Define class of all (non-unital) rings. A non-unital ring (or rng, or pseudoring) is a set equipped with two everywhere-defined internal operations, whose first one is an additive abelian group operation and the second one is a multiplicative semigroup operation, and where the addition is left- and right-distributive for the multiplication. Definition of a pseudo-ring in section I.8.1 of [BourbakiAlg1] p. 93 or the definition of a ring in part Preliminaries of [Roman] p. 18. As almost always in mathematics, "non-unital" means "not necessarily unital". Therefore, by talking about a ring (in general) or a non-unital ring the "unital" case is always included. In contrast to a unital ring, the commutativity of addition must be postulated and cannot be proven from the other conditions. (Contributed by AV, 6-Jan-2020.)
Rng = {𝑓 ∈ Abel ∣ ((mulGrp‘𝑓) ∈ SGrp ∧ [(Base‘𝑓) / 𝑏][(+g𝑓) / 𝑝][(.r𝑓) / 𝑡]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑡(𝑦𝑝𝑧)) = ((𝑥𝑡𝑦)𝑝(𝑥𝑡𝑧)) ∧ ((𝑥𝑝𝑦)𝑡𝑧) = ((𝑥𝑡𝑧)𝑝(𝑦𝑡𝑧))))}

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
 Copyright terms: Public domain < Previous  Next >