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Theorem List for Metamath Proof Explorer - 41601-41700   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremmgmplusgiopALT 41601 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 41602 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 41603 Extend class notation with class of internal (binary) operations for a set.
class intOp

Syntaxcclintop 41604 Extend class notation with class of closed operations for a set.
class clIntOp

Syntaxcassintop 41605 Extend class notation with class of associative operations for a set.
class assIntOp

Definitiondf-intop 41606* 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 41607 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 41608* 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 41609 The internal (binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
((𝑀𝑉𝑁𝑊) → (𝑀 intOp 𝑁) = (𝑁𝑚 (𝑀 × 𝑀)))

Theoremintop 41610 An internal (binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ (𝑀 intOp 𝑁) → :(𝑀 × 𝑀)⟶𝑁)

Theoremclintopval 41611 The closed (internal binary) operations for a set. (Contributed by AV, 20-Jan-2020.)
(𝑀𝑉 → ( clIntOp ‘𝑀) = (𝑀𝑚 (𝑀 × 𝑀)))

Theoremassintopval 41612* 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 41613* 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 41614 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 41615 A closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( clIntOp ‘𝑀) → :(𝑀 × 𝑀)⟶𝑀)

Theoremassintop 41616 An associative (closed internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( assIntOp ‘𝑀) → ( :(𝑀 × 𝑀)⟶𝑀 assLaw 𝑀))

Theoremisassintop 41617* 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 41618 The closure law holds for a closed (internal binary) operation for a set. (Contributed by AV, 20-Jan-2020.)
( ∈ ( clIntOp ‘𝑀) → clLaw 𝑀)

Theoremassintopcllaw 41619 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 41620 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 41621* 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 41622 Extend class notation with class of all magmas.
class MgmALT

Syntaxccmgm2 41623 Extend class notation with class of all commutative magmas.
class CMgmALT

Syntaxcsgrp2 41624 Extend class notation with class of all semigroups.
class SGrpALT

Syntaxccsgrp2 41625 Extend class notation with class of all commutative semigroups.
class CSGrpALT

Definitiondf-mgm2 41626 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 41627 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 41628 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 41629 A commutative semigroup is a semigroup with a commutative operation. (Contributed by AV, 20-Jan-2020.)
CSGrpALT = {𝑔 ∈ SGrpALT ∣ (+g𝑔) comLaw (Base‘𝑔)}

TheoremismgmALT 41630 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 41631 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 41632 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 41633 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 41634 Equivalence of the two definitions of a magma. (Contributed by AV, 16-Jan-2020.)
(𝑀 ∈ MgmALT ↔ 𝑀 ∈ Mgm)

Theoremsgrp2sgrp 41635 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 41636* 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 41637* 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 41638* 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 41639 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 41640 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 41641 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 41642 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 41643 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 41644 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 41645 Extend class notation with class of all non-unital rings.
class Rng

Definitiondf-rng0 41646* 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))))}

Theoremisrng 41647* The predicate "is a non-unital ring." (Contributed by AV, 6-Jan-2020.)
𝐵 = (Base‘𝑅)    &   𝐺 = (mulGrp‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Rng ↔ (𝑅 ∈ Abel ∧ 𝐺 ∈ SGrp ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 · (𝑦 + 𝑧)) = ((𝑥 · 𝑦) + (𝑥 · 𝑧)) ∧ ((𝑥 + 𝑦) · 𝑧) = ((𝑥 · 𝑧) + (𝑦 · 𝑧)))))

Theoremrngabl 41648 A non-unital ring is an (additive) abelian group. (Contributed by AV, 17-Feb-2020.)
(𝑅 ∈ Rng → 𝑅 ∈ Abel)

Theoremrngmgp 41649 A non-unital ring is a semigroup under multiplication. (Contributed by AV, 17-Feb-2020.)
𝐺 = (mulGrp‘𝑅)       (𝑅 ∈ Rng → 𝐺 ∈ SGrp)

Theoremringrng 41650 A unital ring is a (non-unital) ring. (Contributed by AV, 6-Jan-2020.)
(𝑅 ∈ Ring → 𝑅 ∈ Rng)

Theoremringssrng 41651 The unital rings are (non-unital) rings. (Contributed by AV, 20-Mar-2020.)
Ring ⊆ Rng

Theoremisringrng 41652* The predicate "is a unital ring" as extension of the predicate "is a non-unital ring". (Contributed by AV, 17-Feb-2020.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring ↔ (𝑅 ∈ Rng ∧ ∃𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑦 ∧ (𝑦 · 𝑥) = 𝑦)))

Theoremrngdir 41653 Distributive law for the multiplication operation of a nonunital ring (right-distributivity). (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Rng ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 + 𝑌) · 𝑍) = ((𝑋 · 𝑍) + (𝑌 · 𝑍)))

Theoremrngcl 41654 Closure of the multiplication operation of a nonunital ring. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Rng ∧ 𝑋𝐵𝑌𝐵) → (𝑋 · 𝑌) ∈ 𝐵)

Theoremrnglz 41655 The zero of a nonunital ring is a left-absorbing element. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Rng ∧ 𝑋𝐵) → ( 0 · 𝑋) = 0 )

20.35.14.3  Rng homomorphisms

Syntaxcrngh 41656 non-unital ring homomorphisms.
class RngHomo

Syntaxcrngs 41657 non-unital ring isomorphisms.
class RngIsom

Definitiondf-rnghomo 41658* Define the set of non-unital ring homomorphisms from 𝑟 to 𝑠. (Contributed by AV, 20-Feb-2020.)
RngHomo = (𝑟 ∈ Rng, 𝑠 ∈ Rng ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦)))})

Definitiondf-rngisom 41659* Define the set of non-unital ring isomorphisms from 𝑟 to 𝑠. (Contributed by AV, 20-Feb-2020.)
RngIsom = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RngHomo 𝑠) ∣ 𝑓 ∈ (𝑠 RngHomo 𝑟)})

Theoremrnghmrcl 41660 Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020.)
(𝐹 ∈ (𝑅 RngHomo 𝑆) → (𝑅 ∈ Rng ∧ 𝑆 ∈ Rng))

Theoremrnghmfn 41661 The mapping of two non-unital rings to the non-unital ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
RngHomo Fn (Rng × Rng)

Theoremrnghmval 41662* The set of the non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 22-Feb-2020.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    = (.r𝑆)    &   𝐶 = (Base‘𝑆)    &    + = (+g𝑅)    &    = (+g𝑆)       ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ∧ (𝑓‘(𝑥 · 𝑦)) = ((𝑓𝑥) (𝑓𝑦)))})

Theoremisrnghm 41663* A function is a non-unital ring homomorphism iff it is a group homomorphism and preserves multiplication. (Contributed by AV, 22-Feb-2020.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    = (.r𝑆)       (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))

Theoremisrnghmmul 41664 A function is a non-unital ring homomorphism iff it preserves both addition and multiplication. (Contributed by AV, 27-Feb-2020.)
𝑀 = (mulGrp‘𝑅)    &   𝑁 = (mulGrp‘𝑆)       (𝐹 ∈ (𝑅 RngHomo 𝑆) ↔ ((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MgmHom 𝑁))))

Theoremrnghmmgmhm 41665 A non-unital ring homomorphism is a homomorphism of multiplicative magmas. (Contributed by AV, 27-Feb-2020.)
𝑀 = (mulGrp‘𝑅)    &   𝑁 = (mulGrp‘𝑆)       (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹 ∈ (𝑀 MgmHom 𝑁))

Theoremrnghmval2 41666 The non-unital ring homomorphisms between two non-unital rings. (Contributed by AV, 1-Mar-2020.)
((𝑅 ∈ Rng ∧ 𝑆 ∈ Rng) → (𝑅 RngHomo 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MgmHom (mulGrp‘𝑆))))

Theoremisrngisom 41667 An isomorphism of non-unital rings is a homomorphism whose converse is also a homomorphism. (Contributed by AV, 22-Feb-2020.)
((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹 ∈ (𝑆 RngHomo 𝑅))))

Theoremrngimrcl 41668 Reverse closure for an isomorphism of non-unital rings. (Contributed by AV, 22-Feb-2020.)
(𝐹 ∈ (𝑅 RngIsom 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Theoremrnghmghm 41669 A non-unital ring homomorphism is an additive group homomorphism. (Contributed by AV, 23-Feb-2020.)
(𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Theoremrnghmf 41670 A ring homomorphism is a function. (Contributed by AV, 23-Feb-2020.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RngHomo 𝑆) → 𝐹:𝐵𝐶)

Theoremrnghmmul 41671 A homomorphism of non-unital rings preserves multiplication. (Contributed by AV, 23-Feb-2020.)
𝑋 = (Base‘𝑅)    &    · = (.r𝑅)    &    × = (.r𝑆)       ((𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))

Theoremisrnghm2d 41672* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    × = (.r𝑆)    &   (𝜑𝑅 ∈ Rng)    &   (𝜑𝑆 ∈ Rng)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))    &   (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))       (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))

Theoremisrnghmd 41673* Demonstration of non-unital ring homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    × = (.r𝑆)    &   (𝜑𝑅 ∈ Rng)    &   (𝜑𝑆 ∈ Rng)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))    &   𝐶 = (Base‘𝑆)    &    + = (+g𝑅)    &    = (+g𝑆)    &   (𝜑𝐹:𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))       (𝜑𝐹 ∈ (𝑅 RngHomo 𝑆))

Theoremrnghmf1o 41674 A non-unital ring homomorphism is bijective iff its converse is also a non-unital ring homomorphism. (Contributed by AV, 27-Feb-2020.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RngHomo 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 RngHomo 𝑅)))

Theoremisrngim 41675 An isomorphism of non-unital rings is a bijective homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RngIsom 𝑆) ↔ (𝐹 ∈ (𝑅 RngHomo 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))

Theoremrngimf1o 41676 An isomorphism of non-unital rings is a bijection. (Contributed by AV, 23-Feb-2020.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RngIsom 𝑆) → 𝐹:𝐵1-1-onto𝐶)

Theoremrngimrnghm 41677 An isomorphism of non-unital rings is a homomorphism. (Contributed by AV, 23-Feb-2020.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RngIsom 𝑆) → 𝐹 ∈ (𝑅 RngHomo 𝑆))

Theoremrnghmco 41678 The composition of non-unital ring homomorphisms is a homomorphism. (Contributed by AV, 27-Feb-2020.)
((𝐹 ∈ (𝑇 RngHomo 𝑈) ∧ 𝐺 ∈ (𝑆 RngHomo 𝑇)) → (𝐹𝐺) ∈ (𝑆 RngHomo 𝑈))

Theoremidrnghm 41679 The identity homomorphism on a non-unital ring. (Contributed by AV, 27-Feb-2020.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Rng → ( I ↾ 𝐵) ∈ (𝑅 RngHomo 𝑅))

Theoremc0mgm 41680* The constant mapping to zero is a magma homomorphism into a monoid. Remark: Instead of the assumption that T is a monoid, it would be sufficient that T is a magma with a right or left identity. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐻 = (𝑥𝐵0 )       ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MgmHom 𝑇))

Theoremc0mhm 41681* The constant mapping to zero is a monoid homomorphism. (Contributed by AV, 16-Apr-2020.)
𝐵 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐻 = (𝑥𝐵0 )       ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd) → 𝐻 ∈ (𝑆 MndHom 𝑇))

Theoremc0ghm 41682* The constant mapping to zero is a group homomorphism. (Contributed by AV, 16-Apr-2020.)
𝐵 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐻 = (𝑥𝐵0 )       ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → 𝐻 ∈ (𝑆 GrpHom 𝑇))

Theoremc0rhm 41683* The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐻 = (𝑥𝐵0 )       ((𝑆 ∈ Ring ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RingHom 𝑇))

Theoremc0rnghm 41684* The constant mapping to zero is a nonunital ring homomorphism from any nonunital ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑆)    &    0 = (0g𝑇)    &   𝐻 = (𝑥𝐵0 )       ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHomo 𝑇))

Theoremc0snmgmhm 41685* The constant mapping to zero is a magma homomorphism from a magma with one element to any monoid. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑇)    &    0 = (0g𝑆)    &   𝐻 = (𝑥𝐵0 )       ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mgm ∧ (#‘𝐵) = 1) → 𝐻 ∈ (𝑇 MgmHom 𝑆))

Theoremc0snmhm 41686* The constant mapping to zero is a monoid homomorphism from the trivial monoid (consisting of the zero only) to any monoid. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑇)    &    0 = (0g𝑆)    &   𝐻 = (𝑥𝐵0 )    &   𝑍 = (0g𝑇)       ((𝑆 ∈ Mnd ∧ 𝑇 ∈ Mnd ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 MndHom 𝑆))

Theoremc0snghm 41687* The constant mapping to zero is a group homomorphism from the trivial group (consisting of the zero only) to any group. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑇)    &    0 = (0g𝑆)    &   𝐻 = (𝑥𝐵0 )    &   𝑍 = (0g𝑇)       ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐵 = {𝑍}) → 𝐻 ∈ (𝑇 GrpHom 𝑆))

Theoremzrrnghm 41688* The constant mapping to zero is a nonunital ring homomorphism from the zero ring to any nonunital ring. (Contributed by AV, 17-Apr-2020.)
𝐵 = (Base‘𝑇)    &    0 = (0g𝑆)    &   𝐻 = (𝑥𝐵0 )       ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHomo 𝑆))

20.35.14.4  Ring homomorphisms (extension)

Theoremrhmfn 41689 The mapping of two rings to the ring homomorphisms between them is a function. (Contributed by AV, 1-Mar-2020.)
RingHom Fn (Ring × Ring)

Theoremrhmval 41690 The ring homomorphisms between two rings. (Contributed by AV, 1-Mar-2020.)
((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) → (𝑅 RingHom 𝑆) = ((𝑅 GrpHom 𝑆) ∩ ((mulGrp‘𝑅) MndHom (mulGrp‘𝑆))))

Theoremrhmisrnghm 41691 Each unital ring homomorphism is a non-unital ring homomorphism. (Contributed by AV, 29-Feb-2020.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 RngHomo 𝑆))

20.35.14.5  Ideals as non-unital rings

Theoremlidldomn1 41692* 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 ))

Theoremlidlssbas 41693 The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)       (𝑈𝐿 → (Base‘𝐼) ⊆ (Base‘𝑅))

Theoremlidlbas 41694 A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)       (𝑈𝐿 → (Base‘𝐼) = 𝑈)

Theoremlidlabl 41695 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)

Theoremlidlmmgm 41696 The multiplicative group of a (left) ideal of a ring is a magma. (Contributed by AV, 17-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)       ((𝑅 ∈ Ring ∧ 𝑈𝐿) → (mulGrp‘𝐼) ∈ Mgm)

Theoremlidlmsgrp 41697 The multiplicative group of a (left) ideal of a ring is a semigroup. (Contributed by AV, 17-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)       ((𝑅 ∈ Ring ∧ 𝑈𝐿) → (mulGrp‘𝐼) ∈ SGrp)

Theoremlidlrng 41698 A (left) ideal of a ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.)
𝐿 = (LIdeal‘𝑅)    &   𝐼 = (𝑅s 𝑈)       ((𝑅 ∈ Ring ∧ 𝑈𝐿) → 𝐼 ∈ Rng)

Theoremzlidlring 41699 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 41700 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)))

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