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Theorem List for Metamath Proof Explorer - 43001-43100   *Has distinct variable group(s)
TypeLabelDescription
Statement

SyntaxcringcALTV 43001 Extend class notation to include the category Ring. (New usage is discouraged.)
class RingCatALTV

Definitiondf-ringc 43002 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 43003* 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 43004* 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 43005 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 43006 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 43007 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 43008 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 43009 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 43010 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 43011 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 43012 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 43013 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 43014 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 43015 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 43016* 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 43017* 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 43018 Lemma 1 for rhmsubcsetc 43020. (Contributed by AV, 9-Mar-2020.)
𝐶 = (ExtStrCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubcsetclem2 43019* Lemma 2 for rhmsubcsetc 43020. (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 43020 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 43021 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 43022 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 43023 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 43024 Lemma 1 for rhmsubcrngc 43026. (Contributed by AV, 9-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubcrngclem2 43025* Lemma 2 for rhmsubcrngc 43026. (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 43026 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 43027 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 43028 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 43029 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 43030 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 43031 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 43032* 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 43033* Lemma 1 for funcringcsetcALTV2 43042. (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 43034* Lemma 2 for funcringcsetcALTV2 43042. (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 43035* Lemma 3 for funcringcsetcALTV2 43042. (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 43036* Lemma 4 for funcringcsetcALTV2 43042. (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 43037* Lemma 5 for funcringcsetcALTV2 43042. (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 43038* Lemma 6 for funcringcsetcALTV2 43042. (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 43039* Lemma 7 for funcringcsetcALTV2 43042. (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 43040* Lemma 8 for funcringcsetcALTV2 43042. (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 43041* Lemma 9 for funcringcsetcALTV2 43042. (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 43042* 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 43043 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 43044* 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 43045 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 43046 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 43047* 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 43048 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 43049* Lemma for ringccatALTV 43050. (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 43050 The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

TheoremringcidALTV 43051 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 43052 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 43053 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 43054 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 43055 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 43056* Lemma 1 for funcringcsetcALTV 43065. (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 43057* Lemma 2 for funcringcsetcALTV 43065. (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 43058* Lemma 3 for funcringcsetcALTV 43065. (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 43059* Lemma 4 for funcringcsetcALTV 43065. (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 43060* Lemma 5 for funcringcsetcALTV 43065. (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 43061* Lemma 6 for funcringcsetcALTV 43065. (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 43062* Lemma 7 for funcringcsetcALTV 43065. (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 43063* Lemma 8 for funcringcsetcALTV 43065. (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 43064* Lemma 9 for funcringcsetcALTV 43065. (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 43065* 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 43066 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 43067 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 43068* 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 43069 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) = ∅)

20.36.17.10  Subcategories of the category of rings

Theoremsrhmsubclem1 43070* Lemma 1 for srhmsubc 43073. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))

Theoremsrhmsubclem2 43071* Lemma 2 for srhmsubc 43073. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈)))

Theoremsrhmsubclem3 43072* Lemma 3 for srhmsubc 43073. (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))

Theoremsrhmsubc 43073* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

Theoremsringcat 43074* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)

Theoremcrhmsubc 43075* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

Theoremcringcat 43076* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)

Theoremdrhmsubc 43077* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))

Theoremdrngcat 43078* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)

Theoremfldcat 43079* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat)

Theoremfldc 43080* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)

Theoremfldhmsubc 43081* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽)))

Theoremrngcrescrhm 43082 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))

Theoremrhmsubclem1 43083 Lemma 1 for rhmsubc 43087. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))

Theoremrhmsubclem2 43084 Lemma 2 for rhmsubc 43087. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Theoremrhmsubclem3 43085* Lemma 3 for rhmsubc 43087. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubclem4 43086* Lemma 4 for rhmsubc 43087. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))

Theoremrhmsubc 43087 According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 ∈ (Subcat‘(RngCat‘𝑈)))

Theoremrhmsubccat 43088 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → ((RngCat‘𝑈) ↾cat 𝐻) ∈ Cat)

TheoremsrhmsubcALTVlem1 43089* Lemma 1 for srhmsubcALTV 43091. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈)))

TheoremsrhmsubcALTVlem2 43090* Lemma 2 for srhmsubcALTV 43091. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       ((𝑈𝑉 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌))

TheoremsrhmsubcALTV 43091* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of special ring homomorphisms (i.e. ring homomorphisms from a special ring to another ring of that kind) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

TheoremsringcatALTV 43092* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)

TheoremcrhmsubcALTV 43093* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

TheoremcringcatALTV 43094* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)

TheoremdrhmsubcALTV 43095* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))

TheoremdrngcatALTV 43096* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)

TheoremfldcatALTV 43097* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐹) ∈ Cat)

TheoremfldcALTV 43098* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → (((RingCatALTV‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)

TheoremfldhmsubcALTV 43099* According to df-subc 16857, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16885 and subcss2 16888). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)))

TheoremrngcrescrhmALTV 43100 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))

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