P. 483 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 48201-48300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	1odd 48201*	1 is an odd integer. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    ⇒   ⊢ 1 ∈ 𝑂
Theorem	2nodd 48202*	2 is not an odd integer. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    ⇒   ⊢ 2 ∉ 𝑂
Theorem	oddibas 48203*	Lemma 1 for oddinmgm 48205: 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 = (ℂfld ↾s 𝑂)    ⇒   ⊢ 𝑂 = (Base‘𝑀)
Theorem	oddiadd 48204*	Lemma 2 for oddinmgm 48205: 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 = (ℂfld ↾s 𝑂)    ⇒   ⊢ + = (+g‘𝑀)
Theorem	oddinmgm 48205*	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 48280, and even a non-unital ring, see 2zrng 48271. (Contributed by AV, 3-Feb-2020.)
⊢ 𝑂 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = ((2 · 𝑥) + 1)}    &   ⊢ 𝑀 = (ℂfld ↾s 𝑂)    ⇒   ⊢ 𝑀 ∉ Mgm
Theorem	nnsgrpmgm 48206	The structure of positive integers together with the addition of complex numbers is a magma. (Contributed by AV, 4-Feb-2020.)
⊢ 𝑀 = (ℂfld ↾s ℕ)    ⇒   ⊢ 𝑀 ∈ Mgm
Theorem	nnsgrp 48207	The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020.)
⊢ 𝑀 = (ℂfld ↾s ℕ)    ⇒   ⊢ 𝑀 ∈ Smgrp
Theorem	nnsgrpnmnd 48208	The structure of positive integers together with the addition of complex numbers is not a monoid. (Contributed by AV, 4-Feb-2020.)
⊢ 𝑀 = (ℂfld ↾s ℕ)    ⇒   ⊢ 𝑀 ∉ Mnd
Theorem	nn0mnd 48209	The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: 𝑀 could have also been written as (ℂfld ↾s ℕ0). (Contributed by AV, 27-Dec-2023.)
⊢ 𝑀 = {⟨(Base‘ndx), ℕ0⟩, ⟨(+g‘ndx), + ⟩}    ⇒   ⊢ 𝑀 ∈ Mnd
21.48.16.4  Group sum operation (extension 1)
Theorem	gsumsplit2f 48210*	Split a group sum into two parts. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp 0 )    &   ⊢ (𝜑 → (𝐶 ∩ 𝐷) = ∅)    &   ⊢ (𝜑 → 𝐴 = (𝐶 ∪ 𝐷))    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ 𝐶 ↦ 𝑋)) + (𝐺 Σg (𝑘 ∈ 𝐷 ↦ 𝑋))))
Theorem	gsumdifsndf 48211*	Extract a summand from a finitely supported group sum. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑘𝑌    &   ⊢ Ⅎ𝑘𝜑    &   ⊢ 𝐵 = (Base‘𝐺)    &   ⊢ + = (+g‘𝐺)    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑊)    &   ⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝑋) finSupp (0g‘𝐺))    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑀 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝑋 = 𝑌)    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ 𝐴 ↦ 𝑋)) = ((𝐺 Σg (𝑘 ∈ (𝐴 ∖ {𝑀}) ↦ 𝑋)) + 𝑌))
Theorem	gsumfsupp 48212	A group sum of a family can be restricted to the support of that family without changing its value, provided that that support is finite. This corresponds to the definition of an (infinite) product in [Lang] p. 5, last two formulas. (Contributed by AV, 27-Dec-2023.)
⊢ 𝐵 = (Base‘𝐺)    &   ⊢ 0 = (0g‘𝐺)    &   ⊢ 𝐼 = (𝐹 supp 0 )    &   ⊢ (𝜑 → 𝐺 ∈ CMnd)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)    &   ⊢ (𝜑 → 𝐹 finSupp 0 )    ⇒   ⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐼)) = (𝐺 Σg 𝐹))
21.48.17  Magmas and internal binary operations (alternate approach) With df-mpo 7351, binary operations are defined by a rule, and with df-ov 7349, 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 7349 (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, we call binary operations mapping the elements of the Cartesian product 𝑆 × 𝑆 internal binary operations, see df-intop 48229. If, in addition, the result is also contained in the set 𝑆, the operation is called closed internal binary operation, see df-clintop 48230. 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 48230 ). 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 48230 and df-assintop 48231), 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 48216 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 48224, or for an alternate definition df-mgm2 48249 for a magma as extensible structure. Similar results are obtained for an associative operation (defining semigroups).
21.48.17.1  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.
Syntax	ccllaw 48213	Extend class notation for the closure law.
class clLaw
Syntax	casslaw 48214	Extend class notation for the associative law.
class assLaw
Syntax	ccomlaw 48215	Extend class notation for the commutative law.
class comLaw
Definition	df-cllaw 48216*	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⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
Definition	df-comlaw 48217*	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⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 (𝑥𝑜𝑦) = (𝑦𝑜𝑥)}
Definition	df-asslaw 48218*	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⟩ ∣ ∀𝑥 ∈ 𝑚 ∀𝑦 ∈ 𝑚 ∀𝑧 ∈ 𝑚 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
Theorem	iscllaw 48219*	The predicate "is a closed operation". (Contributed by AV, 13-Jan-2020.)
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ clLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) ∈ 𝑀))
Theorem	iscomlaw 48220*	The predicate "is a commutative operation". (Contributed by AV, 20-Jan-2020.)
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ comLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 (𝑥 ⚬ 𝑦) = (𝑦 ⚬ 𝑥)))
Theorem	clcllaw 48221	Closure of a closed operation. (Contributed by FL, 14-Sep-2010.) (Revised by AV, 21-Jan-2020.)
⊢ (( ⚬ clLaw 𝑀 ∧ 𝑋 ∈ 𝑀 ∧ 𝑌 ∈ 𝑀) → (𝑋 ⚬ 𝑌) ∈ 𝑀)
Theorem	isasslaw 48222*	The predicate "is an associative operation". (Contributed by FL, 1-Nov-2009.) (Revised by AV, 13-Jan-2020.)
⊢ (( ⚬ ∈ 𝑉 ∧ 𝑀 ∈ 𝑊) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
Theorem	asslawass 48223*	Associativity of an associative operation. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 21-Jan-2020.)
⊢ ( ⚬ assLaw 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
Theorem	mgmplusgiopALT 48224	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 → (+g‘𝑀) clLaw (Base‘𝑀))
Theorem	sgrpplusgaopALT 48225	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 → (+g‘𝐺) assLaw (Base‘𝐺))
21.48.17.2  Internal binary operations In this subsection, "internal binary operations" obeying different laws are defined.
Syntax	cintop 48226	Extend class notation with class of internal (binary) operations for a set.
class intOp
Syntax	cclintop 48227	Extend class notation with class of closed operations for a set.
class clIntOp
Syntax	cassintop 48228	Extend class notation with class of associative operations for a set.
class assIntOp
Definition	df-intop 48229*	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 ↦ (𝑛 ↑m (𝑚 × 𝑚)))
Definition	df-clintop 48230	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))
Definition	df-assintop 48231*	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})
Theorem	intopval 48232	The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ((𝑀 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → (𝑀 intOp 𝑁) = (𝑁 ↑m (𝑀 × 𝑀)))
Theorem	intop 48233	An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ (𝑀 intOp 𝑁) → ⚬ :(𝑀 × 𝑀)⟶𝑁)
Theorem	clintopval 48234	The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( clIntOp ‘𝑀) = (𝑀 ↑m (𝑀 × 𝑀)))
Theorem	assintopval 48235*	The associative (closed internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
Theorem	assintopmap 48236*	The associative (closed internal binary) operations for a set, expressed with set exponentiation. (Contributed by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
Theorem	isclintop 48237	The predicate "is a closed (internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
Theorem	clintop 48238	A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
Theorem	assintop 48239	An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ⚬ assLaw 𝑀))
Theorem	isassintop 48240*	The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)
⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
Theorem	clintopcllaw 48241	The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
⊢ ( ⚬ ∈ ( clIntOp ‘𝑀) → ⚬ clLaw 𝑀)
Theorem	assintopcllaw 48242	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)
Theorem	assintopasslaw 48243	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)
Theorem	assintopass 48244*	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) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
21.48.17.3  Alternative definitions for magmas and semigroups
Syntax	cmgm2 48245	Extend class notation with class of all magmas.
class MgmALT
Syntax	ccmgm2 48246	Extend class notation with class of all commutative magmas.
class CMgmALT
Syntax	csgrp2 48247	Extend class notation with class of all semigroups.
class SGrpALT
Syntax	ccsgrp2 48248	Extend class notation with class of all commutative semigroups.
class CSGrpALT
Definition	df-mgm2 48249	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 ∣ (+g‘𝑚) clLaw (Base‘𝑚)}
Definition	df-cmgm2 48250	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 ∣ (+g‘𝑚) comLaw (Base‘𝑚)}
Definition	df-sgrp2 48251	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 ∣ (+g‘𝑔) assLaw (Base‘𝑔)}
Definition	df-csgrp2 48252	A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
⊢ CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g‘𝑔) comLaw (Base‘𝑔)}
Theorem	ismgmALT 48253	The predicate "is a magma". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ MgmALT ↔ ⚬ clLaw 𝐵))
Theorem	iscmgmALT 48254	The predicate "is a commutative magma". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ CMgmALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ comLaw 𝐵))
Theorem	issgrpALT 48255	The predicate "is a semigroup". (Contributed by AV, 16-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ SGrpALT ↔ (𝑀 ∈ MgmALT ∧ ⚬ assLaw 𝐵))
Theorem	iscsgrpALT 48256	The predicate "is a commutative semigroup". (Contributed by AV, 20-Jan-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ ⚬ = (+g‘𝑀)    ⇒   ⊢ (𝑀 ∈ CSGrpALT ↔ (𝑀 ∈ SGrpALT ∧ ⚬ comLaw 𝐵))
Theorem	mgm2mgm 48257	Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
⊢ (𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)
Theorem	sgrp2sgrp 48258	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)
Theorem	lmod0rng 48259	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) = {(0g‘𝑀)})
Theorem	nzrneg1ne0 48260	The additive inverse of the 1 in a nonzero ring is not zero ( -1 =/= 0 ). (Contributed by AV, 29-Apr-2019.)
⊢ (𝑅 ∈ NzRing → ((invg‘𝑅)‘(1r‘𝑅)) ≠ (0g‘𝑅))
21.48.18.2  Ideals as non-unital rings
Theorem	lidldomn1 48261*	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)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 }) ∧ 𝐼 ∈ 𝑈) → (∀𝑥 ∈ 𝑈 ((𝐼 · 𝑥) = 𝑥 ∧ (𝑥 · 𝐼) = 𝑥) → 𝐼 = 1 ))
Theorem	lidlabl 48262	A (left) ideal of a ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
⊢ 𝐿 = (LIdeal‘𝑅)    &   ⊢ 𝐼 = (𝑅 ↾s 𝑈)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Abel)
Theorem	lidlrng 48263	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 = (𝑅 ↾s 𝑈)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝐿) → 𝐼 ∈ Rng)
Theorem	zlidlring 48264	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 = (𝑅 ↾s 𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 = { 0 }) → 𝐼 ∈ Ring)
Theorem	uzlidlring 48265	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 = (𝑅 ↾s 𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ 𝑈 ∈ 𝐿) → (𝐼 ∈ Ring ↔ (𝑈 = { 0 } ∨ 𝑈 = 𝐵)))
Theorem	lidldomnnring 48266	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 = (𝑅 ↾s 𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ (𝑈 ∈ 𝐿 ∧ 𝑈 ≠ { 0 } ∧ 𝑈 ≠ 𝐵)) → 𝐼 ∉ Ring)
21.48.18.3  The non-unital ring of even integers
Theorem	0even 48267*	0 is an even integer. (Contributed by AV, 11-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    ⇒   ⊢ 0 ∈ 𝐸
Theorem	1neven 48268*	1 is not an even integer. (Contributed by AV, 12-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    ⇒   ⊢ 1 ∉ 𝐸
Theorem	2even 48269*	2 is an even integer. (Contributed by AV, 12-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    ⇒   ⊢ 2 ∈ 𝐸
Theorem	2zlidl 48270*	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)    ⇒   ⊢ 𝐸 ∈ 𝑈
Theorem	2zrng 48271*	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 48205. (Contributed by AV, 20-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑈 = (LIdeal‘ℤring)    &   ⊢ 𝑅 = (ℤring ↾s 𝐸)    ⇒   ⊢ 𝑅 ∈ Rng
Theorem	2zrngbas 48272*	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 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 𝐸 = (Base‘𝑅)
Theorem	2zrngadd 48273*	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 = (ℂfld ↾s 𝐸)    ⇒   ⊢ + = (+g‘𝑅)
Theorem	2zrng0 48274*	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 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 0 = (0g‘𝑅)
Theorem	2zrngamgm 48275*	R is an (additive) magma. (Contributed by AV, 6-Jan-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 𝑅 ∈ Mgm
Theorem	2zrngasgrp 48276*	R is an (additive) semigroup. (Contributed by AV, 4-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 𝑅 ∈ Smgrp
Theorem	2zrngamnd 48277*	R is an (additive) monoid. (Contributed by AV, 11-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 𝑅 ∈ Mnd
Theorem	2zrngacmnd 48278*	R is a commutative (additive) monoid. (Contributed by AV, 11-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 𝑅 ∈ CMnd
Theorem	2zrngagrp 48279*	R is an (additive) group. (Contributed by AV, 6-Jan-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 𝑅 ∈ Grp
Theorem	2zrngaabl 48280*	R is an (additive) abelian group. (Contributed by AV, 11-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    ⇒   ⊢ 𝑅 ∈ Abel
Theorem	2zrngmul 48281*	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 = (ℂfld ↾s 𝐸)    ⇒   ⊢ · = (.r‘𝑅)
Theorem	2zrngmmgm 48282*	R is a (multiplicative) magma. (Contributed by AV, 11-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ 𝑀 ∈ Mgm
Theorem	2zrngmsgrp 48283*	R is a (multiplicative) semigroup. (Contributed by AV, 4-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ 𝑀 ∈ Smgrp
Theorem	2zrngALT 48284*	The ring of integers restricted to the even integers is a non-unital ring, the "ring of even integers". Alternate version of 2zrng 48271, 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 48280) and a multiplicative semigroup (see 2zrngmsgrp 48283). (Contributed by AV, 11-Feb-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ 𝑅 ∈ Rng
Theorem	2zrngnmlid 48285*	R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ ∀𝑏 ∈ 𝐸 ∃𝑎 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎
Theorem	2zrngnmrid 48286*	R has no multiplicative (right) identity. (Contributed by AV, 12-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑎 · 𝑏) ≠ 𝑎
Theorem	2zrngnmlid2 48287*	R has no multiplicative (left) identity. (Contributed by AV, 12-Feb-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ ∀𝑎 ∈ (𝐸 ∖ {0})∀𝑏 ∈ 𝐸 (𝑏 · 𝑎) ≠ 𝑎
Theorem	2zrngnring 48288*	R is not a unital ring. (Contributed by AV, 6-Jan-2020.)
⊢ 𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   ⊢ 𝑅 = (ℂfld ↾s 𝐸)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ 𝑅 ∉ Ring
21.48.18.4  A constructed not unital ring
Theorem	cznrnglem 48289	Lemma for cznrng 48291: The base set of the ring constructed from a ℤ/nℤ structure by replacing the (multiplicative) ring operation by a constant operation is the base set of the ℤ/nℤ structure. (Contributed by AV, 16-Feb-2020.)
⊢ 𝑌 = (ℤ/nℤ‘𝑁)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑋 = (𝑌 sSet ⟨(.r‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    ⇒   ⊢ 𝐵 = (Base‘𝑋)
Theorem	cznabel 48290	The ring constructed from a ℤ/nℤ 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), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 ∈ 𝐵) → 𝑋 ∈ Abel)
Theorem	cznrng 48291*	The ring constructed from a ℤ/nℤ 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), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    &   ⊢ 0 = (0g‘𝑌)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐶 = 0 ) → 𝑋 ∈ Rng)
Theorem	cznnring 48292*	The ring constructed from a ℤ/nℤ 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), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ 𝐶)⟩)    &   ⊢ 0 = (0g‘𝑌)    ⇒   ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝐶 ∈ 𝐵) → 𝑋 ∉ Ring)
21.48.18.5  The category of non-unital rings (alternate definition) As an alternative to df-rngc 20530, 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 20541.
Syntax	crngcALTV 48293	Extend class notation to include the category Rng. (New usage is discouraged.)
class RngCatALTV
Definition	df-rngcALTV 48294*	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), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RngHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))⟩})
Theorem	rngcvalALTV 48295*	Value of the category of non-unital rings (in a universe). (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))    &   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))    &   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	rngcbasALTV 48296	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
Theorem	rngchomfvalALTV 48297*	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHom 𝑦)))
Theorem	rngchomALTV 48298	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHom 𝑌))
Theorem	elrngchomALTV 48299	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)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
Theorem	rngccofvalALTV 48300*	Composition in the category of non-unital rings. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCatALTV‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓))))