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Theorem List for Metamath Proof Explorer - 42501-42600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrhmsubcsetclem2 42501* Lemma 2 for rhmsubcsetc 42502. (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 42502 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 42503 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 42504 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 42505 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 42506 Lemma 1 for rhmsubcrngc 42508. (Contributed by AV, 9-Mar-2020.)
𝐶 = (RngCat‘𝑈)    &   (𝜑𝑈𝑉)    &   (𝜑𝐵 = (Ring ∩ 𝑈))    &   (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))       ((𝜑𝑥𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubcrngclem2 42507* Lemma 2 for rhmsubcrngc 42508. (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 42508 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 42509 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 42510 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 42511 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 42512 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 42513 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 42514* 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 42515* Lemma 1 for funcringcsetcALTV2 42524. (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 42516* Lemma 2 for funcringcsetcALTV2 42524. (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 42517* Lemma 3 for funcringcsetcALTV2 42524. (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 42518* Lemma 4 for funcringcsetcALTV2 42524. (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 42519* Lemma 5 for funcringcsetcALTV2 42524. (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 42520* Lemma 6 for funcringcsetcALTV2 42524. (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 42521* Lemma 7 for funcringcsetcALTV2 42524. (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 42522* Lemma 8 for funcringcsetcALTV2 42524. (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 42523* Lemma 9 for funcringcsetcALTV2 42524. (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 42524* 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 42525 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 42526* 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 42527 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 42528 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 42529* 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 42530 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 42531* Lemma for ringccatALTV 42532. (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 42532 The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

TheoremringcidALTV 42533 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 42534 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 42535 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 42536 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 42537 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 42538* Lemma 1 for funcringcsetcALTV 42547. (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 42539* Lemma 2 for funcringcsetcALTV 42547. (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 42540* Lemma 3 for funcringcsetcALTV 42547. (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 42541* Lemma 4 for funcringcsetcALTV 42547. (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 42542* Lemma 5 for funcringcsetcALTV 42547. (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 42543* Lemma 6 for funcringcsetcALTV 42547. (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 42544* Lemma 7 for funcringcsetcALTV 42547. (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 42545* Lemma 8 for funcringcsetcALTV 42547. (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 42546* Lemma 9 for funcringcsetcALTV 42547. (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 42547* 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 42548 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 42549 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 42550* 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 42551 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 42552* Lemma 1 for srhmsubc 42555. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))

Theoremsrhmsubclem2 42553* Lemma 2 for srhmsubc 42555. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈)))

Theoremsrhmsubclem3 42554* Lemma 3 for srhmsubc 42555. (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 42555* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42556* 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 42557* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42558* 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 42559* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42560* 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 42561* 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 42562* 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 42563* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42564 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 42565 Lemma 1 for rhmsubc 42569. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))

Theoremrhmsubclem2 42566 Lemma 2 for rhmsubc 42569. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Theoremrhmsubclem3 42567* Lemma 3 for rhmsubc 42569. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubclem4 42568* Lemma 4 for rhmsubc 42569. (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 42569 According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42570 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 42571* Lemma 1 for srhmsubcALTV 42573. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈)))

TheoremsrhmsubcALTVlem2 42572* Lemma 2 for srhmsubcALTV 42573. (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 42573* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42574* 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 42575* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42576* 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 42577* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42578* 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 42579* 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 42580* 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 42581* According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). 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 42582 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⟩))

TheoremrhmsubcALTVlem1 42583 Lemma 1 for rhmsubcALTV 42587. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))

TheoremrhmsubcALTVlem2 42584 Lemma 2 for rhmsubcALTV 42587. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

TheoremrhmsubcALTVlem3 42585* Lemma 3 for rhmsubcALTV 42587. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))

TheoremrhmsubcALTVlem4 42586* Lemma 4 for rhmsubcALTV 42587. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))

TheoremrhmsubcALTV 42587 According to df-subc 16644, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 ( see subcssc 16672 and subcss2 16675). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈)))

TheoremrhmsubcALTVcat 42588 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.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → ((RngCatALTV‘𝑈) ↾cat 𝐻) ∈ Cat)

20.35.15  Basic algebraic structures (extension)

20.35.15.1  Auxiliary theorems

Theoremxpprsng 42589 The Cartesian product of an unordered pair and a singleton. (Contributed by AV, 20-May-2019.)
((𝐴𝑉𝐵𝑊𝐶𝑈) → ({𝐴, 𝐵} × {𝐶}) = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐶⟩})

Theoremopeliun2xp 42590 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5315. (Contributed by AV, 30-Mar-2019.)
(⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))

Theoremeliunxp2 42591* Membership in a union of Cartesian products over its second component, analogous to eliunxp 5403. (Contributed by AV, 30-Mar-2019.)
(𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))

Theoremmpt2mptx2 42592* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐴(𝑦) is not assumed to be constant w.r.t 𝑦, analogous to mpt2mptx 6904. (Contributed by AV, 30-Mar-2019.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Theoremcbvmpt2x2 42593* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpt2 6887 allows 𝐴 to be a function of 𝑦, analogous to cbvmpt2x 6886. (Contributed by AV, 30-Mar-2019.)
𝑧𝐴    &   𝑦𝐷    &   𝑧𝐶    &   𝑤𝐶    &   𝑥𝐸    &   𝑦𝐸    &   (𝑦 = 𝑧𝐴 = 𝐷)    &   ((𝑦 = 𝑧𝑥 = 𝑤) → 𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤𝐷, 𝑧𝐵𝐸)

Theoremdmmpt2ssx2 42594* The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpt2ssx 7391. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})

Theoremmpt2exxg2 42595* Existence of an operation class abstraction (version for dependent domains, i.e. the first base class may depend on the second base class), analogous to mpt2exxg 7400. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)

Theoremovmpt2rdxf 42596* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6939. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)    &   𝑥𝜑    &   𝑦𝜑    &   𝑦𝐴    &   𝑥𝐵    &   𝑥𝑆    &   𝑦𝑆       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2rdx 42597* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpt2dxf 6939. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)

Theoremovmpt2x2 42598* The value of an operation class abstraction. Variant of ovmpt2ga 6943 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝑦 = 𝐵𝐶 = 𝐿)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐿𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)

Theoremfdmdifeqresdif 42599* The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
𝐹 = (𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)))       (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))

Theoremoffvalfv 42600* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))

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