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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ringcid 45001 | 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 45002 | 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 45003 | Lemma 1 for rhmsubcrngc 45005. (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 45004* | Lemma 2 for rhmsubcrngc 45005. (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 45005 | 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 45006 | 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 45007 | 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 45008 | 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 45009 | 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 45010 | 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 45011* | 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 45012* | Lemma 1 for funcringcsetcALTV2 45021. (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 45013* | Lemma 2 for funcringcsetcALTV2 45021. (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 45014* | Lemma 3 for funcringcsetcALTV2 45021. (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 45015* | Lemma 4 for funcringcsetcALTV2 45021. (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 45016* | Lemma 5 for funcringcsetcALTV2 45021. (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 45017* | Lemma 6 for funcringcsetcALTV2 45021. (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 45018* | Lemma 7 for funcringcsetcALTV2 45021. (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 45019* | Lemma 8 for funcringcsetcALTV2 45021. (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 45020* | Lemma 9 for funcringcsetcALTV2 45021. (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 45021* | 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 45022 | 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 45023* | 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 45024 | 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 45025 | 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 45026* | 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 45027 | 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 45028* | Lemma for ringccatALTV 45029. (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 45029 | 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 45030 | 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 45031 | 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 45032 | 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 45033 | 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 45034 | 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 45035* | Lemma 1 for funcringcsetcALTV 45044. (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 45036* | Lemma 2 for funcringcsetcALTV 45044. (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 45037* | Lemma 3 for funcringcsetcALTV 45044. (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 45038* | Lemma 4 for funcringcsetcALTV 45044. (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 45039* | Lemma 5 for funcringcsetcALTV 45044. (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 45040* | Lemma 6 for funcringcsetcALTV 45044. (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 45041* | Lemma 7 for funcringcsetcALTV 45044. (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 45042* | Lemma 8 for funcringcsetcALTV 45044. (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 45043* | Lemma 9 for funcringcsetcALTV 45044. (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 45044* | 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 45045 | 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 45046 | 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 45047* | 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 45048 | 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 45049* | Lemma 1 for srhmsubc 45052. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring)) | ||
Theorem | srhmsubclem2 45050* | Lemma 2 for srhmsubc 45052. (Contributed by AV, 19-Feb-2020.) |
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring & ⊢ 𝐶 = (𝑈 ∩ 𝑆) ⇒ ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈))) | ||
Theorem | srhmsubclem3 45051* | Lemma 3 for srhmsubc 45052. (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 45052* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45053* | 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 45054* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45055* | 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 45056* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45057* | 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 45058* | 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 45059* | 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 45060* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45061 | 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 45062 | Lemma 1 for rhmsubc 45066. (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 45063 | Lemma 2 for rhmsubc 45066. (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 45064* | Lemma 3 for rhmsubc 45066. (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 45065* | Lemma 4 for rhmsubc 45066. (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 45066 | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45067 | 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 45068* | Lemma 1 for srhmsubcALTV 45070. (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 45069* | Lemma 2 for srhmsubcALTV 45070. (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 45070* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45071* | 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 45072* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45073* | 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 45074* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45075* | 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 45076* | 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 45077* | 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 45078* | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45079 | 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 45080 | Lemma 1 for rhmsubcALTV 45084. (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 45081 | Lemma 2 for rhmsubcALTV 45084. (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 45082* | Lemma 3 for rhmsubcALTV 45084. (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 45083* | Lemma 4 for rhmsubcALTV 45084. (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 45084 | According to df-subc 17126, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17154 and subcss2 17157). 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 45085 | 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) | ||
Theorem | opeliun2xp 45086 | Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5581. (Contributed by AV, 30-Mar-2019.) |
⊢ (〈𝐶, 𝑦〉 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ (𝑦 ∈ 𝐵 ∧ 𝐶 ∈ 𝐴)) | ||
Theorem | eliunxp2 45087* | Membership in a union of Cartesian products over its second component, analogous to eliunxp 5670. (Contributed by AV, 30-Mar-2019.) |
⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | ||
Theorem | mpomptx2 45088* | 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 mpomptx 7252. (Contributed by AV, 30-Mar-2019.) |
⊢ (𝑧 = 〈𝑥, 𝑦〉 → 𝐶 = 𝐷) ⇒ ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷) | ||
Theorem | cbvmpox2 45089* | Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7235 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7234. (Contributed by AV, 30-Mar-2019.) |
⊢ Ⅎ𝑧𝐴 & ⊢ Ⅎ𝑦𝐷 & ⊢ Ⅎ𝑧𝐶 & ⊢ Ⅎ𝑤𝐶 & ⊢ Ⅎ𝑥𝐸 & ⊢ Ⅎ𝑦𝐸 & ⊢ (𝑦 = 𝑧 → 𝐴 = 𝐷) & ⊢ ((𝑦 = 𝑧 ∧ 𝑥 = 𝑤) → 𝐶 = 𝐸) ⇒ ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ 𝐷, 𝑧 ∈ 𝐵 ↦ 𝐸) | ||
Theorem | dmmpossx2 45090* | The domain of a mapping is a subset of its base classes expressed as union of Cartesian products over its second component, analogous to dmmpossx 7761. (Contributed by AV, 30-Mar-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) | ||
Theorem | mpoexxg2 45091* | 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 mpoexxg 7771. (Contributed by AV, 30-Mar-2019.) |
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) ⇒ ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V) | ||
Theorem | ovmpordxf 45092* | Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7288. (Contributed by AV, 30-Mar-2019.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) & ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝐿) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) & ⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑦𝜑 & ⊢ Ⅎ𝑦𝐴 & ⊢ Ⅎ𝑥𝐵 & ⊢ Ⅎ𝑥𝑆 & ⊢ Ⅎ𝑦𝑆 ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpordx 45093* | Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7288. (Contributed by AV, 30-Mar-2019.) |
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) & ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) & ⊢ ((𝜑 ∧ 𝑦 = 𝐵) → 𝐶 = 𝐿) & ⊢ (𝜑 → 𝐴 ∈ 𝐿) & ⊢ (𝜑 → 𝐵 ∈ 𝐷) & ⊢ (𝜑 → 𝑆 ∈ 𝑋) ⇒ ⊢ (𝜑 → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | ovmpox2 45094* | The value of an operation class abstraction. Variant of ovmpoga 7292 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 = 𝐴 ∧ 𝑦 = 𝐵) → 𝑅 = 𝑆) & ⊢ (𝑦 = 𝐵 → 𝐶 = 𝐿) & ⊢ 𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) ⇒ ⊢ ((𝐴 ∈ 𝐿 ∧ 𝐵 ∈ 𝐷 ∧ 𝑆 ∈ 𝐻) → (𝐴𝐹𝐵) = 𝑆) | ||
Theorem | fdmdifeqresdif 45095* | 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 = (𝑥 ∈ 𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺‘𝑥))) ⇒ ⊢ (𝐺:(𝐷 ∖ {𝑌})⟶𝑅 → 𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌}))) | ||
Theorem | offvalfv 45096* | The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.) |
⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐹 Fn 𝐴) & ⊢ (𝜑 → 𝐺 Fn 𝐴) ⇒ ⊢ (𝜑 → (𝐹 ∘f 𝑅𝐺) = (𝑥 ∈ 𝐴 ↦ ((𝐹‘𝑥)𝑅(𝐺‘𝑥)))) | ||
Theorem | ofaddmndmap 45097 | The function operation applied to the addition for functions (with the same domain) into a monoid is a function (with the same domain) into the monoid. (Contributed by AV, 6-Apr-2019.) |
⊢ 𝑅 = (Base‘𝑀) & ⊢ + = (+g‘𝑀) ⇒ ⊢ ((𝑀 ∈ Mnd ∧ 𝑉 ∈ 𝑌 ∧ (𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐵 ∈ (𝑅 ↑m 𝑉))) → (𝐴 ∘f + 𝐵) ∈ (𝑅 ↑m 𝑉)) | ||
Theorem | mapsnop 45098 | A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.) |
⊢ 𝐹 = {〈𝑋, 𝑌〉} ⇒ ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑅 ∧ 𝑅 ∈ 𝑊) → 𝐹 ∈ (𝑅 ↑m {𝑋})) | ||
Theorem | fprmappr 45099 | A function with a domain of two elements as element of the mapping operator applied to a pair. (Contributed by AV, 20-May-2024.) |
⊢ ((𝑋 ∈ 𝑉 ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑊 ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ 𝑋 ∧ 𝐷 ∈ 𝑋)) → {〈𝐴, 𝐶〉, 〈𝐵, 𝐷〉} ∈ (𝑋 ↑m {𝐴, 𝐵})) | ||
Theorem | mapprop 45100 | An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.) (Proof shortened by AV, 2-Jun-2024.) |
⊢ 𝐹 = {〈𝑋, 𝐴〉, 〈𝑌, 𝐵〉} ⇒ ⊢ (((𝑋 ∈ 𝑉 ∧ 𝐴 ∈ 𝑅) ∧ (𝑌 ∈ 𝑉 ∧ 𝐵 ∈ 𝑅) ∧ (𝑋 ≠ 𝑌 ∧ 𝑅 ∈ 𝑊)) → 𝐹 ∈ (𝑅 ↑m {𝑋, 𝑌})) |
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