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Theorem List for Metamath Proof Explorer - 42301-42400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrnghmsubcsetclem2 42301* Lemma 2 for rnghmsubcsetc 42302. (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 42302 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 42303 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 42304 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 42305 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 42306 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 42307 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 42308 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 42309* 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 42310 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 42311 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 42312* 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 42313 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 42314* Lemma for rngccatALTV 42315. (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 42315 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 42316 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 42317 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 42318 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 42319 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 42320* 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 42321 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 42322* 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 42323* 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 42324, using cofuval2 16594 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 42322, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16836. (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 42324* Alternate proof of funcrngcsetc 42323, using cofuval2 16594 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 42322, and the "natural forgetful functor" from the category of extensible structures into the category of sets, see funcestrcsetc 16836. Surprisingly, this proof is longer than the direct proof given in funcrngcsetc 42323. (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 42325 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 42326 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 42327 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))

20.35.14.9  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 42330. Alternately, 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 42343. 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 42336, and the morphisms/arrows are the ring homomorphisms restricted to this subset of the rings ( RingHom ↾ (𝐵 × 𝐵)), see ringchomfval 42337, whereas the composition is the ordinary composition of functions, see ringccofval 42341 and ringcco 42342.

By showing that the ring homomorphisms between rings are a subcategory subset (cat) of the mappings between base sets of extensible structures, see rhmsscmap 42345, it can be shown that the ring homomorphisms between rings are a subcategory (Subcat) of the category of extensible structures, see rhmsubcsetc 42348. It follows that the category of rings RingCat is actually a category, see ringccat 42349 with the identity function as identity arrow, see ringcid 42350.

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 42351, and that the ring homomorphisms between rings are a subcategory of the category of non-unital rings, see rhmsubcrngc 42354. 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 42355: ((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 42360.

Syntaxcringc 42328 Extend class notation to include the category Ring.
class RingCat

SyntaxcringcALTV 42329 Extend class notation to include the category Ring. (New usage is discouraged.)
class RingCatALTV

Definitiondf-ringc 42330 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 42331* 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 42332* 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 42333 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 42334 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 42335 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 42336 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 42337 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 42338 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 42339 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 42340 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 42341 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 42342 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 42343 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 42344* 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 42345* 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 42346 Lemma 1 for rhmsubcsetc 42348. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubcsetclem2 42347* Lemma 2 for rhmsubcsetc 42348. (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 42348 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 42349 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 42350 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 42351 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 42352 Lemma 1 for rhmsubcrngc 42354. (Contributed by AV, 9-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubcrngclem2 42353* Lemma 2 for rhmsubcrngc 42354. (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 42354 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 42355 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 42356 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 42357 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 42358 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 42359 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 42360* 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)

TheoremfuncringcsetcALTV2lem1 42361* Lemma 1 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))

TheoremfuncringcsetcALTV2lem2 42362* Lemma 2 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)

TheoremfuncringcsetcALTV2lem3 42363* Lemma 3 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       (𝜑𝐹:𝐵𝐶)

TheoremfuncringcsetcALTV2lem4 42364* Lemma 4 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐺 Fn (𝐵 × 𝐵))

TheoremfuncringcsetcALTV2lem5 42365* Lemma 5 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))

TheoremfuncringcsetcALTV2lem6 42366* Lemma 6 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

TheoremfuncringcsetcALTV2lem7 42367* Lemma 7 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))

TheoremfuncringcsetcALTV2lem8 42368* Lemma 8 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))

TheoremfuncringcsetcALTV2lem9 42369* Lemma 9 for funcringcsetcALTV2 42370. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))

TheoremfuncringcsetcALTV2 42370* 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, 16-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)

TheoremringcbasALTV 42371 Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑𝐵 = (𝑈 ∩ Ring))

TheoremringchomfvalALTV 42372* Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))

TheoremringchomALTV 42373 Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

TheoremelringchomALTV 42374 A morphism of rings is a function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))

TheoremringccofvalALTV 42375* Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))

TheoremringccoALTV 42376 Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋 RingHom 𝑌))    &   (𝜑𝐺 ∈ (𝑌 RingHom 𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))

TheoremringccatidALTV 42377* Lemma for ringccatALTV 42378. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))

TheoremringccatALTV 42378 The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

TheoremringcidALTV 42379 The identity arrow in the category of rings is the identity function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   𝑆 = (Base‘𝑋)       (𝜑 → ( 1𝑋) = ( I ↾ 𝑆))

TheoremringcsectALTV 42380 A section in the category of rings, written out. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘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))))

TheoremringcinvALTV 42381 An inverse in the category of rings is the converse operation. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))

TheoremringcisoALTV 42382 An isomorphism in the category of rings is a bijection. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))

TheoremringcbasbasALTV 42383 An element of the base set of the base set of the category of rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈 ∈ WUni)       ((𝜑𝑅𝐵) → (Base‘𝑅) ∈ 𝑈)

Theoremfuncringcsetclem1ALTV 42384* Lemma 1 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))

Theoremfuncringcsetclem2ALTV 42385* Lemma 2 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)

Theoremfuncringcsetclem3ALTV 42386* Lemma 3 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       (𝜑𝐹:𝐵𝐶)

Theoremfuncringcsetclem4ALTV 42387* Lemma 4 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐺 Fn (𝐵 × 𝐵))

Theoremfuncringcsetclem5ALTV 42388* Lemma 5 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))

Theoremfuncringcsetclem6ALTV 42389* Lemma 6 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Theoremfuncringcsetclem7ALTV 42390* Lemma 7 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))

Theoremfuncringcsetclem8ALTV 42391* Lemma 8 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))

Theoremfuncringcsetclem9ALTV 42392* Lemma 9 for funcringcsetcALTV 42393. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))

TheoremfuncringcsetcALTV 42393* The "natural forgetful functor" from the category of 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, 16-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)

Theoremirinitoringc 42394 The ring of integers is an initial object in the category of unital rings (within a universe containing the ring of integers). Example 7.2 (6) of [Adamek] p. 101 , and example in [Lang] p. 58. (Contributed by AV, 3-Apr-2020.)
(𝜑𝑈𝑉)    &   (𝜑 → ℤring𝑈)    &   𝐶 = (RingCat‘𝑈)       (𝜑 → ℤring ∈ (InitO‘𝐶))

Theoremzrtermoringc 42395 The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RingCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)       (𝜑𝑍 ∈ (TermO‘𝐶))

Theoremzrninitoringc 42396* The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RingCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)    &   (𝜑 → ∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing)       (𝜑𝑍 ∉ (InitO‘𝐶))

Theoremnzerooringczr 42397 There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RingCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)    &   (𝜑 → ℤring𝑈)       (𝜑 → (ZeroO‘𝐶) = ∅)

20.35.14.10  Subcategories of the category of rings

Theoremsrhmsubclem1 42398* Lemma 1 for srhmsubc 42401. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))

Theoremsrhmsubclem2 42399* Lemma 2 for srhmsubc 42401. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈)))

Theoremsrhmsubclem3 42400* Lemma 3 for srhmsubc 42401. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       ((𝑈𝑉 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCat‘𝑈))𝑌))

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