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Theorem List for Metamath Proof Explorer - 48101-48200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxccomlaw 48101 Extend class notation for the commutative law.
class comLaw
 
Definitiondf-cllaw 48102* 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 48103* 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 48104* 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 48105* The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)
(( 𝑉𝑀𝑊) → ( clLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) ∈ 𝑀))
 
Theoremiscomlaw 48106* The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.)
(( 𝑉𝑀𝑊) → ( comLaw 𝑀 ↔ ∀𝑥𝑀𝑦𝑀 (𝑥 𝑦) = (𝑦 𝑥)))
 
Theoremclcllaw 48107 Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)
(( clLaw 𝑀𝑋𝑀𝑌𝑀) → (𝑋 𝑌) ∈ 𝑀)
 
Theoremisasslaw 48108* 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 48109* 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 48110 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 48111 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 ∈ Smgrp → (+g𝐺) assLaw (Base‘𝐺))
 
21.48.17.2  Internal binary operations

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

 
Syntaxcintop 48112 Extend class notation with class of internal (binary) operations for a set.
class intOp
 
Syntaxcclintop 48113 Extend class notation with class of closed operations for a set.
class clIntOp
 
Syntaxcassintop 48114 Extend class notation with class of associative operations for a set.
class assIntOp
 
Definitiondf-intop 48115* 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 48116 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 48117* 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 48118 The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁m (𝑀 × 𝑀)))
 
Theoremintop 48119 An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ (𝑀 intOp 𝑁) → :(𝑀 × 𝑀)⟶𝑁)
 
Theoremclintopval 48120 The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
(𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀m (𝑀 × 𝑀)))
 
Theoremassintopval 48121* 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 48122* 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 48123 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 48124 A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)
 
Theoremassintop 48125 An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))
 
Theoremisassintop 48126* 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 48127 The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)
 
Theoremassintopcllaw 48128 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 48129 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 48130* 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)))
 
21.48.17.3  Alternative definitions for magmas and semigroups
 
Syntaxcmgm2 48131 Extend class notation with class of all magmas.
class MgmALT
 
Syntaxccmgm2 48132 Extend class notation with class of all commutative magmas.
class CMgmALT
 
Syntaxcsgrp2 48133 Extend class notation with class of all semigroups.
class SGrpALT
 
Syntaxccsgrp2 48134 Extend class notation with class of all commutative semigroups.
class CSGrpALT
 
Definitiondf-mgm2 48135 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 48136 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 48137 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 semigroup in section 1.3 of [Hall] p. 7. (Contributed by AV, 6-Jan-2020.)
SGrpALT = {𝑔 ∈ MgmALT ∣ (+g𝑔) assLaw (Base‘𝑔)}
 
Definitiondf-csgrp2 48138 A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g𝑔) comLaw (Base‘𝑔)}
 
TheoremismgmALT 48139 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 48140 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 48141 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 48142 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 48143 Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
(𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
 
Theoremsgrp2sgrp 48144 Equivalence of the two definitions of a semigroup. (Contributed by AV, 16-Jan-2020.)
(𝑀 ∈ SGrpALT ↔ 𝑀 ∈ Smgrp)
 
21.48.18  Rings (extension)
 
21.48.18.1  Nonzero rings (extension)
 
Theoremlmod0rng 48145 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 48146 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))
 
21.48.18.2  Ideals as non-unital rings
 
Theoremlidldomn1 48147* If a (left) ideal (which is not the zero ideal) of a domain has a multiplicative identity element, the identity element is the identity of the domain. (Contributed by AV, 17-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ (𝑈𝐿𝑈 ≠ { 0 }) ∧ 𝐼𝑈) → (∀𝑥𝑈 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥) → 𝐼 = 1 ))
 
Theoremlidlabl 48148 A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)       ((𝑅 ∈ Ring ∧ 𝑈𝐿) → 𝐼 ∈ Abel)
 
Theoremlidlrng 48149 A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) (Proof shortened by AV, 11-Mar-2025.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)       ((𝑅 ∈ Ring ∧ 𝑈𝐿) → 𝐼 ∈ Rng)
 
Theoremzlidlring 48150 The zero (left) ideal of a non-unital ring is a unital ring (the zero ring). (Contributed by AV, 16-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
 
Theoremuzlidlring 48151 Only the zero (left) ideal or the unit (left) ideal of a domain is a unital ring. (Contributed by AV, 18-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ 𝑈𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
 
Theoremlidldomnnring 48152 A (left) ideal of a domain which is neither the zero ideal nor the unit ideal is not a unital ring. (Contributed by AV, 18-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ (𝑈𝐿𝑈 ≠ { 0 } ∧ 𝑈𝐵)) → 𝐼 ∉ Ring)
 
21.48.18.3  The non-unital ring of even integers
 
Theorem0even 48153* 0 is an even integer. (Contributed by AV, 11-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}       0 ∈ 𝐸
 
Theorem1neven 48154* 1 is not an even integer. (Contributed by AV, 12-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}       1 ∉ 𝐸
 
Theorem2even 48155* 2 is an even integer. (Contributed by AV, 12-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}       2 ∈ 𝐸
 
Theorem2zlidl 48156* The even integers are a (left) ideal of the ring of integers. (Contributed by AV, 20-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑈 = (LIdeal‘ℤring)       𝐸𝑈
 
Theorem2zrng 48157* The ring of integers restricted to the even integers is a non-unital ring, the "ring of even integers". Remark: the structure of the complementary subset of the set of integers, the odd integers, is not even a magma, see oddinmgm 48091. (Contributed by AV, 20-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑈 = (LIdeal‘ℤring)    &   𝑅 = (ℤrings 𝐸)       𝑅 ∈ Rng
 
Theorem2zrngbas 48158* The base set of R is the set of all even integers. (Contributed by AV, 31-Jan-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝐸 = (Base‘𝑅)
 
Theorem2zrngadd 48159* The group addition operation of R is the addition of complex numbers. (Contributed by AV, 31-Jan-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)        + = (+g𝑅)
 
Theorem2zrng0 48160* The additive identity of R is the complex number 0. (Contributed by AV, 11-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       0 = (0g𝑅)
 
Theorem2zrngamgm 48161* R is an (additive) magma. (Contributed by AV, 6-Jan-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝑅 ∈ Mgm
 
Theorem2zrngasgrp 48162* R is an (additive) semigroup. (Contributed by AV, 4-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝑅 ∈ Smgrp
 
Theorem2zrngamnd 48163* R is an (additive) monoid. (Contributed by AV, 11-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝑅 ∈ Mnd
 
Theorem2zrngacmnd 48164* R is a commutative (additive) monoid. (Contributed by AV, 11-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝑅 ∈ CMnd
 
Theorem2zrngagrp 48165* R is an (additive) group. (Contributed by AV, 6-Jan-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝑅 ∈ Grp
 
Theorem2zrngaabl 48166* R is an (additive) abelian group. (Contributed by AV, 11-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝑅 ∈ Abel
 
Theorem2zrngmul 48167* The ring multiplication operation of R is the multiplication on complex numbers. (Contributed by AV, 31-Jan-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)        · = (.r𝑅)
 
Theorem2zrngmmgm 48168* R is a (multiplicative) magma. (Contributed by AV, 11-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)    &   𝑀 = (mulGrp‘𝑅)       𝑀 ∈ Mgm
 
Theorem2zrngmsgrp 48169* R is a (multiplicative) semigroup. (Contributed by AV, 4-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)    &   𝑀 = (mulGrp‘𝑅)       𝑀 ∈ Smgrp
 
Theorem2zrngALT 48170* The ring of integers restricted to the even integers is a non-unital ring, the "ring of even integers". Alternate version of 2zrng 48157, based on a restriction of the field of the complex numbers. The proof is based on the facts that the ring of even integers is an additive abelian group (see 2zrngaabl 48166) and a multiplicative semigroup (see 2zrngmsgrp 48169). (Contributed by AV, 11-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)    &   𝑀 = (mulGrp‘𝑅)       𝑅 ∈ Rng
 
Theorem2zrngnmlid 48171* R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)    &   𝑀 = (mulGrp‘𝑅)       𝑏𝐸𝑎𝐸 (𝑏 · 𝑎) ≠ 𝑎
 
Theorem2zrngnmrid 48172* R has no multiplicative (right) identity. (Contributed by AV, 12-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)    &   𝑀 = (mulGrp‘𝑅)       𝑎 ∈ (𝐸 ∖ {0})∀𝑏𝐸 (𝑎 · 𝑏) ≠ 𝑎
 
Theorem2zrngnmlid2 48173* R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)    &   𝑀 = (mulGrp‘𝑅)       𝑎 ∈ (𝐸 ∖ {0})∀𝑏𝐸 (𝑏 · 𝑎) ≠ 𝑎
 
Theorem2zrngnring 48174* R is not a unital ring. (Contributed by AV, 6-Jan-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)    &   𝑀 = (mulGrp‘𝑅)       𝑅 ∉ Ring
 
21.48.18.4  A constructed not unital ring
 
Theoremcznrnglem 48175 Lemma for cznrng 48177: The base set of the ring constructed from a ℤ/n structure by replacing the (multiplicative) ring operation by a constant operation is the base set of the ℤ/n structure. (Contributed by AV, 16-Feb-2020.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)    &   𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵𝐶)⟩)       𝐵 = (Base‘𝑋)
 
Theoremcznabel 48176 The ring constructed from a ℤ/n structure by replacing the (multiplicative) ring operation by a constant operation is an abelian group. (Contributed by AV, 16-Feb-2020.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)    &   𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵𝐶)⟩)       ((𝑁 ∈ ℕ ∧ 𝐶𝐵) → 𝑋 ∈ Abel)
 
Theoremcznrng 48177* The ring constructed from a ℤ/n structure by replacing the (multiplicative) ring operation by a constant operation is a non-unital ring. (Contributed by AV, 17-Feb-2020.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)    &   𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵𝐶)⟩)    &    0 = (0g𝑌)       ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑋 ∈ Rng)
 
Theoremcznnring 48178* The ring constructed from a ℤ/n structure with 1 < 𝑛 by replacing the (multiplicative) ring operation by a constant operation is not a unital ring. (Contributed by AV, 17-Feb-2020.)
𝑌 = (ℤ/nℤ‘𝑁)    &   𝐵 = (Base‘𝑌)    &   𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥𝐵, 𝑦𝐵𝐶)⟩)    &    0 = (0g𝑌)       ((𝑁 ∈ (ℤ‘2) ∧ 𝐶𝐵) → 𝑋 ∉ Ring)
 
21.48.18.5  The category of non-unital rings (alternate definition)

As an alternative to df-rngc 20617, the "category of non-unital rings" can be defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, according to dfrngc2 20628.

 
SyntaxcrngcALTV 48179 Extend class notation to include the category Rng. (New usage is discouraged.)
class RngCatALTV
 
Definitiondf-rngcALTV 48180* Definition of the category Rng, relativized to a subset 𝑢. This is the category of all non-unital rings in 𝑢 and homomorphisms between these rings. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
RngCatALTV = (𝑢 ∈ V ↦ (𝑢 ∩ Rng) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHom 𝑧), 𝑓 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑔𝑓)))⟩})
 
TheoremrngcvalALTV 48181* Value of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Rng))    &   (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)))    &   (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHom 𝑧), 𝑓 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑔𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
 
TheoremrngcbasALTV 48182 Set of objects of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑𝐵 = (𝑈 ∩ Rng))
 
TheoremrngchomfvalALTV 48183* Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)))
 
TheoremrngchomALTV 48184 Set of arrows of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHom 𝑌))
 
TheoremelrngchomALTV 48185 A morphism of non-unital rings is a function. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
 
TheoremrngccofvalALTV 48186* Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHom 𝑧), 𝑓 ∈ ((1st𝑣) RngHom (2nd𝑣)) ↦ (𝑔𝑓))))
 
TheoremrngccoALTV 48187 Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋 RngHom 𝑌))    &   (𝜑𝐺 ∈ (𝑌 RngHom 𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
 
TheoremrngccatidALTV 48188* Lemma for rngccatALTV 48189. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
 
TheoremrngccatALTV 48189 The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.)
𝐶 = (RngCatALTV‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)
 
TheoremrngcidALTV 48190 The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   𝑆 = (Base‘𝑋)       (𝜑 → ( 1𝑋) = ( I ↾ 𝑆))
 
TheoremrngcsectALTV 48191 A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐸 = (Base‘𝑋)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHom 𝑌) ∧ 𝐺 ∈ (𝑌 RngHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
 
TheoremrngcinvALTV 48192 An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIso 𝑌) ∧ 𝐺 = 𝐹)))
 
TheoremrngcisoALTV 48193 An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.) (New usage is discouraged.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIso 𝑌)))
 
TheoremrngchomffvalALTV 48194* The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) in maps-to notation for an operation. (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐹 = (Homf𝐶)       (𝜑𝐹 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RngHom 𝑦)))
 
TheoremrngchomrnghmresALTV 48195 The value of the functionalized Hom-set operation in the category of non-unital rings (in a universe) as restriction of the non-unital ring homomorphisms. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
𝐶 = (RngCatALTV‘𝑈)    &   𝐵 = (Rng ∩ 𝑈)    &   (𝜑𝑈𝑉)    &   𝐹 = (Homf𝐶)       (𝜑𝐹 = ( RngHom ↾ (𝐵 × 𝐵)))
 
TheoremrngcrescrhmALTV 48196 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))
 
TheoremrhmsubcALTVlem1 48197 Lemma 1 for rhmsubcALTV 48201. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))
 
TheoremrhmsubcALTVlem2 48198 Lemma 2 for rhmsubcALTV 48201. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
 
TheoremrhmsubcALTVlem3 48199* Lemma 3 for rhmsubcALTV 48201. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
 
TheoremrhmsubcALTVlem4 48200* Lemma 4 for rhmsubcALTV 48201. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
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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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48500 486 48501-48600 487 48601-48700 488 48701-48800 489 48801-48900 490 48901-49000 491 49001-49100 492 49101-49200 493 49201-49300 494 49301-49324
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