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Theorem List for Metamath Proof Explorer - 41701-41800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2zrngbas 41701* 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 41702* 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 41703* 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 41704* 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 41705* R is an (additive) semigroup. (Contributed by AV, 4-Feb-2020.)
𝐸 = {𝑧 ∈ ℤ ∣ ∃𝑥 ∈ ℤ 𝑧 = (2 · 𝑥)}    &   𝑅 = (ℂflds 𝐸)       𝑅 ∈ SGrp

Theorem2zrngamnd 41706* 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 41707* 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 41708* 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 41709* 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 41710* 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 41711* 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 41712* 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 ∈ SGrp

Theorem2zrngALT 41713* The ring of integers restricted to the even integers is a (non-unital) ring, the "ring of even integers". Alternate version of 2zrng 41700, 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 41709) and a multiplicative semigroup (see 2zrngmsgrp 41712). (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 41714* 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 41715* 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 41716* 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 41717* 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

20.34.14.7  A constructed not unital ring

Theoremcznrnglem 41718 Lemma for cznrng 41720: 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 41719 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 41720* 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 41721* 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)

20.34.14.8  The category of non-unital rings

The "category of non-unital rings" RngCat is the category of all non-unital rings Rng in a universe and non-unital ring homomorphisms RngHomo between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to non-unital rings and the morphisms to the non-unital ring homomorphisms, while the composition of morphisms is preserved, see df-rngc 41724. Alternatively, the category of non-unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see df-rngcALTV 41725 or dfrngc2 41737.

Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are a subset of the non-unital rings (relativized to a subset or "universe" 𝑢) (𝑢 ∩ Rng), see rngcbas 41730, and the morphisms/arrows are the non-unital ring homomorphisms restricted to this subset of the non-unital rings ( RngHomo ↾ (𝐵 × 𝐵)), see rngchomfval 41731, whereas the composition is the ordinary composition of functions, see rngccofval 41735 and rngcco 41736.

By showing that the non-unital ring homomorphisms between non-unital rings are a subcategory subset (cat) of the mappings between base sets of extensible structures, see rnghmsscmap 41739, it can be shown that the non-unital ring homomorphisms between non-unital rings are a subcategory (Subcat) of the category of extensible structures, see rnghmsubcsetc 41742. It follows that the category of non-unital rings RngCat is actually a category, see rngccat 41743 with the identity function as identity arrow, see rngcid 41744.

Syntaxcrngc 41722 Extend class notation to include the category Rng.
class RngCat

SyntaxcrngcALTV 41723 Extend class notation to include the category Rng. (New usage is discouraged.)
class RngCatALTV

Definitiondf-rngc 41724 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. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))

Definitiondf-rngcALTV 41725* 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 RngHomo 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓)))⟩})

TheoremrngcvalALTV 41726* 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 RngHomo 𝑦)))    &   (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremrngcval 41727 Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Rng))    &   (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))       (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Theoremrnghmresfn 41728 The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020.)
(𝜑𝐵 = (𝑈 ∩ Rng))    &   (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))       (𝜑𝐻 Fn (𝐵 × 𝐵))

Theoremrnghmresel 41729 An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.)
(𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHomo 𝑌))

Theoremrngcbas 41730 Set of objects of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑𝐵 = (𝑈 ∩ Rng))

Theoremrngchomfval 41731 Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))

Theoremrngchom 41732 Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌))

Theoremelrngchom 41733 A morphism of non-unital rings is a function. (Contributed by AV, 27-Feb-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))

Theoremrngchomfeqhom 41734 The functionalized Hom-set operation equals the Hom-set operation in the category of non-unital rings (in a universe). (Contributed by AV, 9-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑 → (Homf𝐶) = (Hom ‘𝐶))

Theoremrngccofval 41735 Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (comp‘(ExtStrCat‘𝑈)))

Theoremrngcco 41736 Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑈)    &   (𝜑𝐹:(Base‘𝑋)⟶(Base‘𝑌))    &   (𝜑𝐺:(Base‘𝑌)⟶(Base‘𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Theoremdfrngc2 41737 Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Rng))    &   (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))    &   (𝜑· = (comp‘(ExtStrCat‘𝑈)))       (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremrnghmsscmap2 41738* The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
(𝜑𝑈𝑉)    &   (𝜑𝑅 = (Rng ∩ 𝑈))       (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))

Theoremrnghmsscmap 41739* The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)
(𝜑𝑈𝑉)    &   (𝜑𝑅 = (Rng ∩ 𝑈))       (𝜑 → ( RngHomo ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))

Theoremrnghmsubcsetclem1 41740 Lemma 1 for rnghmsubcsetc 41742. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Rng ∩ 𝑈))    &   (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrnghmsubcsetclem2 41741* Lemma 2 for rnghmsubcsetc 41742. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Rng ∩ 𝑈))    &   (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))

Theoremrnghmsubcsetc 41742 The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Rng ∩ 𝑈))    &   (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))       (𝜑𝐻 ∈ (Subcat‘𝐶))

Theoremrngccat 41743 The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 9-Mar-2020.)
𝐶 = (RngCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremrngcid 41744 The identity arrow in the category of non-unital rings is the identity function. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 10-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   𝑆 = (Base‘𝑋)       (𝜑 → ( 1𝑋) = ( I ↾ 𝑆))

Theoremrngcsect 41745 A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐸 = (Base‘𝑋)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))

Theoremrngcinv 41746 An inverse in the category of non-unital rings is the converse operation. (Contributed by AV, 28-Feb-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = 𝐹)))

Theoremrngciso 41747 An isomorphism in the category of non-unital rings is a bijection. (Contributed by AV, 28-Feb-2020.)
𝐶 = (RngCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIsom 𝑌)))

TheoremrngcbasALTV 41748 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 41749* 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 RngHomo 𝑦)))

TheoremrngchomALTV 41750 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 RngHomo 𝑌))

TheoremelrngchomALTV 41751 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 41752* 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) RngHomo 𝑧), 𝑓 ∈ ((1st𝑣) RngHomo (2nd𝑣)) ↦ (𝑔𝑓))))

TheoremrngccoALTV 41753 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 RngHomo 𝑌))    &   (𝜑𝐺 ∈ (𝑌 RngHomo 𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

TheoremrngccatidALTV 41754* Lemma for rngccatALTV 41755. (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 41755 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 41756 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 41757 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 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))

TheoremrngcinvALTV 41758 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 RngIsom 𝑌) ∧ 𝐺 = 𝐹)))

TheoremrngcisoALTV 41759 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 RngIsom 𝑌)))

TheoremrngchomffvalALTV 41760* 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 RngHomo 𝑦)))

TheoremrngchomrnghmresALTV 41761 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 = ( RngHomo ↾ (𝐵 × 𝐵)))

Theoremrngcifuestrc 41762* The "inclusion functor" from the category of non-unital rings into the category of extensible structures. (Contributed by AV, 30-Mar-2020.)
𝑅 = (RngCat‘𝑈)    &   𝐸 = (ExtStrCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑈𝑉)    &   (𝜑𝐹 = ( I ↾ 𝐵))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))       (𝜑𝐹(𝑅 Func 𝐸)𝐺)

Theoremfuncrngcsetc 41763* The "natural forgetful functor" from the category of non-unital rings into the category of sets which sends each non-unital ring to its underlying set (base set) and the morphisms (non-unital ring homomorphisms) to mappings of the corresponding base sets. An alternate proof is provided in funcrngcsetcALT 41764, using cofuval2 16528 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 41762, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16770. (Contributed by AV, 26-Mar-2020.)
𝑅 = (RngCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)

TheoremfuncrngcsetcALT 41764* Alternate proof of funcrngcsetc 41763, using cofuval2 16528 to construct the "natural forgetful functor" from the category of non-unital rings into the category of sets by composing the "inclusion functor" from the category of non-unital rings into the category of extensible structures, see rngcifuestrc 41762, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16770. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 41763. (Contributed by AV, 30-Mar-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑅 = (RngCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)

Theoremzrinitorngc 41765 The zero ring is an initial object in the category of nonunital rings. (Contributed by AV, 18-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)       (𝜑𝑍 ∈ (InitO‘𝐶))

Theoremzrtermorngc 41766 The zero ring is a terminal object in the category of nonunital rings. (Contributed by AV, 17-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)       (𝜑𝑍 ∈ (TermO‘𝐶))

Theoremzrzeroorngc 41767 The zero ring is a zero object in the category of non-unital rings. (Contributed by AV, 18-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)       (𝜑𝑍 ∈ (ZeroO‘𝐶))

20.34.14.9  The category of (unital) rings

The "category of unital rings" RingCat is the category of all (unital) rings Ring in a universe and (unital) ring homomorphisms RingHom between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to (unital) rings and the morphisms to the (unital) ring homomorphisms, while the composition of morphisms is preserved, see df-ringc 41770. Alternatively, the category of unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see dfringc2 41783. In the following, we omit the predicate "unital", so that "ring" and "ring homomorphism" (without predicate) always mean "unital ring" and "unital ring homomorphism".

Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are a subset of the rings (relativized to a subset or "universe"𝑢) (𝑢 ∩ Ring), see ringcbas 41776, and the morphisms/arrows are the ring homomorphisms restricted to this subset of the rings ( RingHom ↾ (𝐵 × 𝐵)), see ringchomfval 41777, whereas the composition is the ordinary composition of functions, see ringccofval 41781 and ringcco 41782.

By showing that the ring homomorphisms between rings are a subcategory subset (cat) of the mappings between base sets of extensible structures, see rhmsscmap 41785, it can be shown that the ring homomorphisms between rings are a subcategory (Subcat) of the category of extensible structures, see rhmsubcsetc 41788. It follows that the category of rings RingCat is actually a category, see ringccat 41789 with the identity function as identity arrow, see ringcid 41790.

Furthermore, it is shown that the ring homomorphisms between rings are a subcategory subset of the non-unital ring homomorphisms between non-unital rings, see rhmsscrnghm 41791, and that the ring homomorphisms between rings are a subcategory of the category of non-unital rings, see rhmsubcrngc 41794. By this, the restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings, see rngcresringcat 41795: ((RngCat‘𝑈) ↾cat ( RingHom ↾ (𝐵 × 𝐵))) = (RingCat‘𝑈)).

Finally, it is shown that the "natural forgetful functor" from the category of rings into the category of sets is the function which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets, see funcringcsetc 41800.

Syntaxcringc 41768 Extend class notation to include the category Ring.
class RingCat

SyntaxcringcALTV 41769 Extend class notation to include the category Ring. (New usage is discouraged.)
class RingCatALTV

Definitiondf-ringc 41770 Definition of the category Ring, relativized to a subset 𝑢. See also the note in [Lang] p. 91, and the item Rng in [Adamek] p. 478. This is the category of all 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. (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))

Definitiondf-ringcALTV 41771* Definition of the category Ring, relativized to a subset 𝑢. This is the category of all 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. (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
RingCatALTV = (𝑢 ∈ V ↦ (𝑢 ∩ Ring) / 𝑏{⟨(Base‘ndx), 𝑏⟩, ⟨(Hom ‘ndx), (𝑥𝑏, 𝑦𝑏 ↦ (𝑥 RingHom 𝑦))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧𝑏 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓)))⟩})

TheoremringcvalALTV 41772* Value of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Ring))    &   (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))    &   (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))       (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremringcval 41773 Value of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Ring))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Theoremrhmresfn 41774 The class of unital ring homomorphisms restricted to subsets of unital rings is a function. (Contributed by AV, 10-Mar-2020.)
(𝜑𝐵 = (𝑈 ∩ Ring))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       (𝜑𝐻 Fn (𝐵 × 𝐵))

Theoremrhmresel 41775 An element of the unital ring homomorphisms restricted to a subset of unital rings is a unital ring homomorphism. (Contributed by AV, 10-Mar-2020.)
(𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌))

Theoremringcbas 41776 Set of objects of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑𝐵 = (𝑈 ∩ Ring))

Theoremringchomfval 41777 Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))

Theoremringchom 41778 Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Theoremelringchom 41779 A morphism of unital rings is a function. (Contributed by AV, 14-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))

Theoremringchomfeqhom 41780 The functionalized Hom-set operation equals the Hom-set operation in the category of unital rings (in a universe). (Contributed by AV, 9-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑 → (Homf𝐶) = (Hom ‘𝐶))

Theoremringccofval 41781 Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (comp‘(ExtStrCat‘𝑈)))

Theoremringcco 41782 Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)    &   (𝜑𝑍𝑈)    &   (𝜑𝐹:(Base‘𝑋)⟶(Base‘𝑌))    &   (𝜑𝐺:(Base‘𝑌)⟶(Base‘𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

Theoremdfringc2 41783 Alternate definition of the category of unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (𝑈 ∩ Ring))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    &   (𝜑· = (comp‘(ExtStrCat‘𝑈)))       (𝜑𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})

Theoremrhmsscmap2 41784* The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
(𝜑𝑈𝑉)    &   (𝜑𝑅 = (Ring ∩ 𝑈))       (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑅, 𝑦𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))

Theoremrhmsscmap 41785* The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)
(𝜑𝑈𝑉)    &   (𝜑𝑅 = (Ring ∩ 𝑈))       (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥𝑈, 𝑦𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥))))

Theoremrhmsubcsetclem1 41786 Lemma 1 for rhmsubcsetc 41788. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubcsetclem2 41787* Lemma 2 for rhmsubcsetc 41788. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))

Theoremrhmsubcsetc 41788 The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       (𝜑𝐻 ∈ (Subcat‘𝐶))

Theoremringccat 41789 The category of unital rings is a category. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 9-Mar-2020.)
𝐶 = (RingCat‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

Theoremringcid 41790 The identity arrow in the category of unital rings is the identity function. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 10-Mar-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   𝑆 = (Base‘𝑋)       (𝜑 → ( 1𝑋) = ( I ↾ 𝑆))

Theoremrhmsscrnghm 41791 The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the non-unital ring homomorphisms between non-unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
(𝜑𝑈𝑉)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   (𝜑𝑆 = (Rng ∩ 𝑈))       (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆)))

Theoremrhmsubcrngclem1 41792 Lemma 1 for rhmsubcrngc 41794. (Contributed by AV, 9-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubcrngclem2 41793* Lemma 2 for rhmsubcrngc 41794. (Contributed by AV, 12-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ∀𝑦𝐵𝑧𝐵𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))

Theoremrhmsubcrngc 41794 The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of non-unital rings. (Contributed by AV, 12-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       (𝜑𝐻 ∈ (Subcat‘𝐶))

Theoremrngcresringcat 41795 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       (𝜑 → (𝐶cat 𝐻) = (RingCat‘𝑈))

Theoremringcsect 41796 A section in the category of unital rings, written out. (Contributed by AV, 14-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐸 = (Base‘𝑋)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))

Theoremringcinv 41797 An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))

Theoremringciso 41798 An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))

Theoremringcbasbas 41799 An element of the base set of the base set of the category of unital rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈 ∈ WUni)       ((𝜑𝑅𝐵) → (Base‘𝑅) ∈ 𝑈)

Theoremfuncringcsetc 41800* The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)

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