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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | rhmresel 44301 | 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 44302 | 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 44303 | 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 44304 | 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 44305 | 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 44306 | 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 44307 | 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 44308 | 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 44309 | 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 44310* | 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 44311* | 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 44312 | Lemma 1 for rhmsubcsetc 44314. (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 44313* | Lemma 2 for rhmsubcsetc 44314. (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 44314 | 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 44315 | 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 44316 | 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 44317 | 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 44318 | Lemma 1 for rhmsubcrngc 44320. (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 44319* | Lemma 2 for rhmsubcrngc 44320. (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 44320 | 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 44321 | 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 44322 | 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 44323 | 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 44324 | 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 44325 | 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 44326* | 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 44327* | Lemma 1 for funcringcsetcALTV2 44336. (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 44328* | Lemma 2 for funcringcsetcALTV2 44336. (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 44329* | Lemma 3 for funcringcsetcALTV2 44336. (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 44330* | Lemma 4 for funcringcsetcALTV2 44336. (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 44331* | Lemma 5 for funcringcsetcALTV2 44336. (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 44332* | Lemma 6 for funcringcsetcALTV2 44336. (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 44333* | Lemma 7 for funcringcsetcALTV2 44336. (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 44334* | Lemma 8 for funcringcsetcALTV2 44336. (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 44335* | Lemma 9 for funcringcsetcALTV2 44336. (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 44336* | 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 44337 | 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 44338* | 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 44339 | 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 44340 | 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 44341* | 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 44342 | 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 44343* | Lemma for ringccatALTV 44344. (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 44344 | 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 44345 | 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 44346 | 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 44347 | 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 44348 | 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 44349 | 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 44350* | Lemma 1 for funcringcsetcALTV 44359. (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 44351* | Lemma 2 for funcringcsetcALTV 44359. (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 44352* | Lemma 3 for funcringcsetcALTV 44359. (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 44353* | Lemma 4 for funcringcsetcALTV 44359. (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 44354* | Lemma 5 for funcringcsetcALTV 44359. (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 44355* | Lemma 6 for funcringcsetcALTV 44359. (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 44356* | Lemma 7 for funcringcsetcALTV 44359. (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 44357* | Lemma 8 for funcringcsetcALTV 44359. (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 44358* | Lemma 9 for funcringcsetcALTV 44359. (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 44359* | 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 44360 | 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 44361 | 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 44362* | 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 44363 | 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 44364* | Lemma 1 for srhmsubc 44367. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) | ||
Theorem | srhmsubclem2 44365* | Lemma 2 for srhmsubc 44367. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈))) | ||
Theorem | srhmsubclem3 44366* | Lemma 3 for srhmsubc 44367. (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)) | ||
Theorem | srhmsubc 44367* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) | ||
Theorem | sringcat 44368* | The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | crhmsubc 44369* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) | ||
Theorem | cringcat 44370* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | drhmsubc 44371* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈))) | ||
Theorem | drngcat 44372* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | fldcat 44373* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat) | ||
Theorem | fldc 44374* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat) | ||
Theorem | fldhmsubc 44375* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽))) | ||
Theorem | rngcrescrhm 44376 | 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet 〈(Hom ‘ndx), 𝐻〉)) | ||
Theorem | rhmsubclem1 44377 | Lemma 1 for rhmsubc 44381. (Contributed by AV, 2-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) | ||
Theorem | rhmsubclem2 44378 | Lemma 2 for rhmsubc 44381. (Contributed by AV, 2-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) | ||
Theorem | rhmsubclem3 44379* | Lemma 3 for rhmsubc 44381. (Contributed by AV, 2-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rhmsubclem4 44380* | Lemma 4 for rhmsubc 44381. (Contributed by AV, 2-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rhmsubc 44381 | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈))) | ||
Theorem | rhmsubccat 44382 | 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCat‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → ((RngCat‘𝑈) ↾cat 𝐻) ∈ Cat) | ||
Theorem | srhmsubcALTVlem1 44383* | Lemma 1 for srhmsubcALTV 44385. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈))) | ||
Theorem | srhmsubcALTVlem2 44384* | Lemma 2 for srhmsubcALTV 44385. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌)) | ||
Theorem | srhmsubcALTV 44385* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈))) | ||
Theorem | sringcatALTV 44386* | 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.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) & ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | crhmsubcALTV 44387* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈))) | ||
Theorem | cringcatALTV 44388* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | drhmsubcALTV 44389* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈))) | ||
Theorem | drngcatALTV 44390* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat) | ||
Theorem | fldcatALTV 44391* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐹) ∈ Cat) | ||
Theorem | fldcALTV 44392* | 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → (((RingCatALTV‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat) | ||
Theorem | fldhmsubcALTV 44393* | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠)) & ⊢ 𝐷 = (𝑈 ∩ Field) & ⊢ 𝐹 = (𝑟 ∈ 𝐷, 𝑠 ∈ 𝐷 ↦ (𝑟 RingHom 𝑠)) ⇒ ⊢ (𝑈 ∈ 𝑉 → 𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽))) | ||
Theorem | rngcrescrhmALTV 44394 | 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet 〈(Hom ‘ndx), 𝐻〉)) | ||
Theorem | rhmsubcALTVlem1 44395 | Lemma 1 for rhmsubcALTV 44399. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅)) | ||
Theorem | rhmsubcALTVlem2 44396 | Lemma 2 for rhmsubcALTV 44399. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌)) | ||
Theorem | rhmsubcALTVlem3 44397* | Lemma 3 for rhmsubcALTV 44399. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥)) | ||
Theorem | rhmsubcALTVlem4 44398* | Lemma 4 for rhmsubcALTV 44399. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(〈𝑥, 𝑦〉(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧)) | ||
Theorem | rhmsubcALTV 44399 | According to df-subc 17082, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17110 and subcss2 17113). 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈))) | ||
Theorem | rhmsubcALTVcat 44400 | 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.) |
⊢ (𝜑 → 𝑈 ∈ 𝑉) & ⊢ 𝐶 = (RngCatALTV‘𝑈) & ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈)) & ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅)) ⇒ ⊢ (𝜑 → ((RngCatALTV‘𝑈) ↾cat 𝐻) ∈ Cat) |
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