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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rnghmsubcsetclem2 42301* | Lemma 2 for rnghmsubcsetc 42302. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rnghmsubcsetc 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RngHomo ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) | ||
Theorem | rngccat 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) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | rngcid 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑆 = (Base‘𝑋) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) | ||
Theorem | rngcsect 42305 | A section in the category of non-unital rings, written out. (Contributed by AV, 28-Feb-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐸 = (Base‘𝑋) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) | ||
Theorem | rngcinv 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = ◡𝐹))) | ||
Theorem | rngciso 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIsom 𝑌))) | ||
Theorem | rngcbasALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng)) | ||
Theorem | rngchomfvalALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦))) | ||
Theorem | rngchomALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHomo 𝑌)) | ||
Theorem | elrngchomALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))) | ||
Theorem | rngccofvalALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RngHomo 𝑧), 𝑓 ∈ ((1st ‘𝑣) RngHomo (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) | ||
Theorem | rngccoALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋 RngHomo 𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌 RngHomo 𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | rngccatidALTV 42314* | Lemma for rngccatALTV 42315. (New usage is discouraged.) (Contributed by AV, 27-Feb-2020.) |
⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥))))) | ||
Theorem | rngccatALTV 42315 | The category of non-unital rings is a category. (Contributed by AV, 27-Feb-2020.) (New usage is discouraged.) |
⊢ 𝐶 = (RngCatALTV‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | rngcidALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑆 = (Base‘𝑋) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) | ||
Theorem | rngcsectALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐸 = (Base‘𝑋) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngHomo 𝑌) ∧ 𝐺 ∈ (𝑌 RngHomo 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) | ||
Theorem | rngcinvALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RngIsom 𝑌) ∧ 𝐺 = ◡𝐹))) | ||
Theorem | rngcisoALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RngIsom 𝑌))) | ||
Theorem | rngchomffvalALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐹 = (Homf ‘𝐶) ⇒ ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RngHomo 𝑦))) | ||
Theorem | rngchomrnghmresALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐹 = (Homf ‘𝐶) ⇒ ⊢ (𝜑 → 𝐹 = ( RngHomo ↾ (𝐵 × 𝐵))) | ||
Theorem | rngcifuestrc 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 = ( I ↾ 𝐵)) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝐸)𝐺) | ||
Theorem | funcrngcsetc 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | funcrngcsetcALT 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RngHomo 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | zrinitorngc 42325 | The zero ring is an initial object in the category of nonunital rings. (Contributed by AV, 18-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (InitO‘𝐶)) | ||
Theorem | zrtermorngc 42326 | The zero ring is a terminal object in the category of nonunital rings. (Contributed by AV, 17-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶)) | ||
Theorem | zrzeroorngc 42327 | The zero ring is a zero object in the category of non-unital rings. (Contributed by AV, 18-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (ZeroO‘𝐶)) | ||
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. | ||
Syntax | cringc 42328 | Extend class notation to include the category Ring. |
class RingCat | ||
Syntax | cringcALTV 42329 | Extend class notation to include the category Ring. (New usage is discouraged.) |
class RingCatALTV | ||
Definition | df-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))))) | ||
Definition | df-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), (𝑥 ∈ 𝑏, 𝑦 ∈ 𝑏 ↦ (𝑥 RingHom 𝑦))〉, 〈(comp‘ndx), (𝑣 ∈ (𝑏 × 𝑏), 𝑧 ∈ 𝑏 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))〉}) | ||
Theorem | ringcvalALTV 42332* | Value of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.) |
⊢ 𝐶 = (RingCatALTV‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) & ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))) & ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | ringcval 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻)) | ||
Theorem | rhmresfn 42334 | The class of unital ring homomorphisms restricted to subsets of unital rings is a function. (Contributed by AV, 10-Mar-2020.) |
⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵)) | ||
Theorem | rhmresel 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.) |
⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RingHom 𝑌)) | ||
Theorem | ringcbas 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | ||
Theorem | ringchomfval 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) | ||
Theorem | ringchom 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) | ||
Theorem | elringchom 42339 | A morphism of unital rings is a function. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))) | ||
Theorem | ringchomfeqhom 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶)) | ||
Theorem | ringccofval 42341 | Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) | ||
Theorem | ringcco 42342 | Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)) & ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | dfringc2 42343 | Alternate definition of the category of unital rings (in a universe). (Contributed by AV, 16-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈))) ⇒ ⊢ (𝜑 → 𝐶 = {〈(Base‘ndx), 𝐵〉, 〈(Hom ‘ndx), 𝐻〉, 〈(comp‘ndx), · 〉}) | ||
Theorem | rhmsscmap2 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | ||
Theorem | rhmsscmap 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑𝑚 (Base‘𝑥)))) | ||
Theorem | rhmsubcsetclem1 42346 | Lemma 1 for rhmsubcsetc 42348. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rhmsubcsetclem2 42347* | Lemma 2 for rhmsubcsetc 42348. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (ExtStrCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rhmsubcsetc 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) | ||
Theorem | ringccat 42349 | The category of unital rings is a category. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 9-Mar-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | ringcid 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑆 = (Base‘𝑋) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) | ||
Theorem | rhmsscrnghm 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝑆 = (Rng ∩ 𝑈)) ⇒ ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHomo ↾ (𝑆 × 𝑆))) | ||
Theorem | rhmsubcrngclem1 42352 | Lemma 1 for rhmsubcrngc 42354. (Contributed by AV, 9-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rhmsubcrngclem2 42353* | Lemma 2 for rhmsubcrngc 42354. (Contributed by AV, 12-Mar-2020.) |
⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(〈𝑥, 𝑦〉(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rhmsubcrngc 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶)) | ||
Theorem | rngcresringcat 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈)) & ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈)) | ||
Theorem | ringcsect 42356 | A section in the category of unital rings, written out. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐸 = (Base‘𝑋) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) | ||
Theorem | ringcinv 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹))) | ||
Theorem | ringciso 42358 | An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.) |
⊢ 𝐶 = (RingCat‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) | ||
Theorem | ringcbasbas 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) ⇒ ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (Base‘𝑅) ∈ 𝑈) | ||
Theorem | funcringcsetc 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | funcringcsetcALTV2lem1 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) | ||
Theorem | funcringcsetcALTV2lem2 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈) | ||
Theorem | funcringcsetcALTV2lem3 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | ||
Theorem | funcringcsetcALTV2lem4 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) | ||
Theorem | funcringcsetcALTV2lem5 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌))) | ||
Theorem | funcringcsetcALTV2lem6 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻) | ||
Theorem | funcringcsetcALTV2lem7 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) | ||
Theorem | funcringcsetcALTV2lem8 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌))) | ||
Theorem | funcringcsetcALTV2lem9 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(〈𝑋, 𝑌〉(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))) | ||
Theorem | funcringcsetcALTV2 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | ringcbasALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring)) | ||
Theorem | ringchomfvalALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) ⇒ ⊢ (𝜑 → 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ (𝑥 RingHom 𝑦))) | ||
Theorem | ringchomALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) | ||
Theorem | elringchomALTV 42374 | A morphism of rings is a function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
⊢ 𝐶 = (RingCatALTV‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐻 = (Hom ‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))) | ||
Theorem | ringccofvalALTV 42375* | Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
⊢ 𝐶 = (RingCatALTV‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) ⇒ ⊢ (𝜑 → · = (𝑣 ∈ (𝐵 × 𝐵), 𝑧 ∈ 𝐵 ↦ (𝑔 ∈ ((2nd ‘𝑣) RingHom 𝑧), 𝑓 ∈ ((1st ‘𝑣) RingHom (2nd ‘𝑣)) ↦ (𝑔 ∘ 𝑓)))) | ||
Theorem | ringccoALTV 42376 | Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
⊢ 𝐶 = (RingCatALTV‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ · = (comp‘𝐶) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝐹 ∈ (𝑋 RingHom 𝑌)) & ⊢ (𝜑 → 𝐺 ∈ (𝑌 RingHom 𝑍)) ⇒ ⊢ (𝜑 → (𝐺(〈𝑋, 𝑌〉 · 𝑍)𝐹) = (𝐺 ∘ 𝐹)) | ||
Theorem | ringccatidALTV 42377* | Lemma for ringccatALTV 42378. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
⊢ 𝐶 = (RingCatALTV‘𝑈) & ⊢ 𝐵 = (Base‘𝐶) ⇒ ⊢ (𝑈 ∈ 𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥 ∈ 𝐵 ↦ ( I ↾ (Base‘𝑥))))) | ||
Theorem | ringccatALTV 42378 | The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.) |
⊢ 𝐶 = (RingCatALTV‘𝑈) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat) | ||
Theorem | ringcidALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ 𝑆 = (Base‘𝑋) ⇒ ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆)) | ||
Theorem | ringcsectALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐸 = (Base‘𝑋) & ⊢ 𝑆 = (Sect‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸)))) | ||
Theorem | ringcinvALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝑁 = (Inv‘𝐶) ⇒ ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹))) | ||
Theorem | ringcisoALTV 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) & ⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ 𝐼 = (Iso‘𝐶) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌))) | ||
Theorem | ringcbasbasALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) ⇒ ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (Base‘𝑅) ∈ 𝑈) | ||
Theorem | funcringcsetclem1ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) = (Base‘𝑋)) | ||
Theorem | funcringcsetclem2ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (𝐹‘𝑋) ∈ 𝑈) | ||
Theorem | funcringcsetclem3ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) ⇒ ⊢ (𝜑 → 𝐹:𝐵⟶𝐶) | ||
Theorem | funcringcsetclem4ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐺 Fn (𝐵 × 𝐵)) | ||
Theorem | funcringcsetclem5ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌))) | ||
Theorem | funcringcsetclem6ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻) | ||
Theorem | funcringcsetclem7ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹‘𝑋))) | ||
Theorem | funcringcsetclem8ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹‘𝑋)(Hom ‘𝑆)(𝐹‘𝑌))) | ||
Theorem | funcringcsetclem9ALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(〈𝑋, 𝑌〉(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(〈(𝐹‘𝑋), (𝐹‘𝑌)〉(comp‘𝑆)(𝐹‘𝑍))((𝑋𝐺𝑌)‘𝐻))) | ||
Theorem | funcringcsetcALTV 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) & ⊢ (𝜑 → 𝑈 ∈ WUni) & ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥))) & ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦)))) ⇒ ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺) | ||
Theorem | irinitoringc 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ (𝜑 → ℤring ∈ 𝑈) & ⊢ 𝐶 = (RingCat‘𝑈) ⇒ ⊢ (𝜑 → ℤring ∈ (InitO‘𝐶)) | ||
Theorem | zrtermoringc 42395 | The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶)) | ||
Theorem | zrninitoringc 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → ∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing) ⇒ ⊢ (𝜑 → 𝑍 ∉ (InitO‘𝐶)) | ||
Theorem | nzerooringczr 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RingCat‘𝑈) & ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing)) & ⊢ (𝜑 → 𝑍 ∈ 𝑈) & ⊢ (𝜑 → ℤring ∈ 𝑈) ⇒ ⊢ (𝜑 → (ZeroO‘𝐶) = ∅) | ||
Theorem | srhmsubclem1 42398* | Lemma 1 for srhmsubc 42401. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) | ||
Theorem | srhmsubclem2 42399* | Lemma 2 for srhmsubc 42401. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈))) | ||
Theorem | srhmsubclem3 42400* | Lemma 3 for srhmsubc 42401. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCat‘𝑈))𝑌)) |
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