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Theorem List for Metamath Proof Explorer - 45601-45700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremringcinv 45601 An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))
 
Theoremringciso 45602 An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))
 
Theoremringcbasbas 45603 An element of the base set of the base set of the category of unital rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.)
𝐶 = (RingCat‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈 ∈ WUni)       ((𝜑𝑅𝐵) → (Base‘𝑅) ∈ 𝑈)
 
Theoremfuncringcsetc 45604* The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)
 
TheoremfuncringcsetcALTV2lem1 45605* Lemma 1 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
 
TheoremfuncringcsetcALTV2lem2 45606* Lemma 2 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
 
TheoremfuncringcsetcALTV2lem3 45607* Lemma 3 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       (𝜑𝐹:𝐵𝐶)
 
TheoremfuncringcsetcALTV2lem4 45608* Lemma 4 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐺 Fn (𝐵 × 𝐵))
 
TheoremfuncringcsetcALTV2lem5 45609* Lemma 5 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
 
TheoremfuncringcsetcALTV2lem6 45610* Lemma 6 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
 
TheoremfuncringcsetcALTV2lem7 45611* Lemma 7 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
 
TheoremfuncringcsetcALTV2lem8 45612* Lemma 8 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
 
TheoremfuncringcsetcALTV2lem9 45613* Lemma 9 for funcringcsetcALTV2 45614. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
 
TheoremfuncringcsetcALTV2 45614* The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCat‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)
 
TheoremringcbasALTV 45615 Set of objects of the category of rings (in a universe). (Contributed by AV, 13-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)       (𝜑𝐵 = (𝑈 ∩ Ring))
 
TheoremringchomfvalALTV 45616* Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)       (𝜑𝐻 = (𝑥𝐵, 𝑦𝐵 ↦ (𝑥 RingHom 𝑦)))
 
TheoremringchomALTV 45617 Set of arrows of the category of rings (in a universe). (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
 
TheoremelringchomALTV 45618 A morphism of rings is a function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   𝐻 = (Hom ‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹:(Base‘𝑋)⟶(Base‘𝑌)))
 
TheoremringccofvalALTV 45619* Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)       (𝜑· = (𝑣 ∈ (𝐵 × 𝐵), 𝑧𝐵 ↦ (𝑔 ∈ ((2nd𝑣) RingHom 𝑧), 𝑓 ∈ ((1st𝑣) RingHom (2nd𝑣)) ↦ (𝑔𝑓))))
 
TheoremringccoALTV 45620 Composition in the category of rings. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &    · = (comp‘𝐶)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑍𝐵)    &   (𝜑𝐹 ∈ (𝑋 RingHom 𝑌))    &   (𝜑𝐺 ∈ (𝑌 RingHom 𝑍))       (𝜑 → (𝐺(⟨𝑋, 𝑌· 𝑍)𝐹) = (𝐺𝐹))
 
TheoremringccatidALTV 45621* Lemma for ringccatALTV 45622. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)       (𝑈𝑉 → (𝐶 ∈ Cat ∧ (Id‘𝐶) = (𝑥𝐵 ↦ ( I ↾ (Base‘𝑥)))))
 
TheoremringccatALTV 45622 The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)
 
TheoremringcidALTV 45623 The identity arrow in the category of rings is the identity function. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &    1 = (Id‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   𝑆 = (Base‘𝑋)       (𝜑 → ( 1𝑋) = ( I ↾ 𝑆))
 
TheoremringcsectALTV 45624 A section in the category of rings, written out. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐸 = (Base‘𝑋)    &   𝑆 = (Sect‘𝐶)       (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺𝐹) = ( I ↾ 𝐸))))
 
TheoremringcinvALTV 45625 An inverse in the category of rings is the converse operation. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝑁 = (Inv‘𝐶)       (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = 𝐹)))
 
TheoremringcisoALTV 45626 An isomorphism in the category of rings is a bijection. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈𝑉)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   𝐼 = (Iso‘𝐶)       (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))
 
TheoremringcbasbasALTV 45627 An element of the base set of the base set of the category of rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)    &   𝐵 = (Base‘𝐶)    &   (𝜑𝑈 ∈ WUni)       ((𝜑𝑅𝐵) → (Base‘𝑅) ∈ 𝑈)
 
Theoremfuncringcsetclem1ALTV 45628* Lemma 1 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) = (Base‘𝑋))
 
Theoremfuncringcsetclem2ALTV 45629* Lemma 2 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       ((𝜑𝑋𝐵) → (𝐹𝑋) ∈ 𝑈)
 
Theoremfuncringcsetclem3ALTV 45630* Lemma 3 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))       (𝜑𝐹:𝐵𝐶)
 
Theoremfuncringcsetclem4ALTV 45631* Lemma 4 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐺 Fn (𝐵 × 𝐵))
 
Theoremfuncringcsetclem5ALTV 45632* Lemma 5 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌) = ( I ↾ (𝑋 RingHom 𝑌)))
 
Theoremfuncringcsetclem6ALTV 45633* Lemma 6 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵) ∧ 𝐻 ∈ (𝑋 RingHom 𝑌)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)
 
Theoremfuncringcsetclem7ALTV 45634* Lemma 7 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑𝑋𝐵) → ((𝑋𝐺𝑋)‘((Id‘𝑅)‘𝑋)) = ((Id‘𝑆)‘(𝐹𝑋)))
 
Theoremfuncringcsetclem8ALTV 45635* Lemma 8 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐺𝑌):(𝑋(Hom ‘𝑅)𝑌)⟶((𝐹𝑋)(Hom ‘𝑆)(𝐹𝑌)))
 
Theoremfuncringcsetclem9ALTV 45636* Lemma 9 for funcringcsetcALTV 45637. (Contributed by AV, 15-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       ((𝜑 ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝐻 ∈ (𝑋(Hom ‘𝑅)𝑌) ∧ 𝐾 ∈ (𝑌(Hom ‘𝑅)𝑍))) → ((𝑋𝐺𝑍)‘(𝐾(⟨𝑋, 𝑌⟩(comp‘𝑅)𝑍)𝐻)) = (((𝑌𝐺𝑍)‘𝐾)(⟨(𝐹𝑋), (𝐹𝑌)⟩(comp‘𝑆)(𝐹𝑍))((𝑋𝐺𝑌)‘𝐻)))
 
TheoremfuncringcsetcALTV 45637* The "natural forgetful functor" from the category of rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 16-Feb-2020.) (New usage is discouraged.)
𝑅 = (RingCatALTV‘𝑈)    &   𝑆 = (SetCat‘𝑈)    &   𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)    &   (𝜑𝑈 ∈ WUni)    &   (𝜑𝐹 = (𝑥𝐵 ↦ (Base‘𝑥)))    &   (𝜑𝐺 = (𝑥𝐵, 𝑦𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))       (𝜑𝐹(𝑅 Func 𝑆)𝐺)
 
Theoremirinitoringc 45638 The ring of integers is an initial object in the category of unital rings (within a universe containing the ring of integers). Example 7.2 (6) of [Adamek] p. 101 , and example in [Lang] p. 58. (Contributed by AV, 3-Apr-2020.)
(𝜑𝑈𝑉)    &   (𝜑 → ℤring𝑈)    &   𝐶 = (RingCat‘𝑈)       (𝜑 → ℤring ∈ (InitO‘𝐶))
 
Theoremzrtermoringc 45639 The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RingCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)       (𝜑𝑍 ∈ (TermO‘𝐶))
 
Theoremzrninitoringc 45640* The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RingCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)    &   (𝜑 → ∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing)       (𝜑𝑍 ∉ (InitO‘𝐶))
 
Theoremnzerooringczr 45641 There is no zero object in the category of unital rings (at least in a universe which contains the zero ring and the ring of integers). Example 7.9 (3) in [Adamek] p. 103. (Contributed by AV, 18-Apr-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RingCat‘𝑈)    &   (𝜑𝑍 ∈ (Ring ∖ NzRing))    &   (𝜑𝑍𝑈)    &   (𝜑 → ℤring𝑈)       (𝜑 → (ZeroO‘𝐶) = ∅)
 
20.41.19.10  Subcategories of the category of rings
 
Theoremsrhmsubclem1 45642* Lemma 1 for srhmsubc 45645. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))
 
Theoremsrhmsubclem2 45643* Lemma 2 for srhmsubc 45645. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈)))
 
Theoremsrhmsubclem3 45644* Lemma 3 for srhmsubc 45645. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       ((𝑈𝑉 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCat‘𝑈))𝑌))
 
Theoremsrhmsubc 45645* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of special ring homomorphisms (i.e., ring homomorphisms from a special ring to another ring of that kind) is a subcategory of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
 
Theoremsringcat 45646* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)
 
Theoremcrhmsubc 45647* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
 
Theoremcringcat 45648* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)
 
Theoremdrhmsubc 45649* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
 
Theoremdrngcat 45650* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)
 
Theoremfldcat 45651* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCat‘𝑈) ↾cat 𝐹) ∈ Cat)
 
Theoremfldc 45652* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → (((RingCat‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)
 
Theoremfldhmsubc 45653* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐹 ∈ (Subcat‘((RingCat‘𝑈) ↾cat 𝐽)))
 
Theoremrngcrescrhm 45654 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))
 
Theoremrhmsubclem1 45655 Lemma 1 for rhmsubc 45659. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))
 
Theoremrhmsubclem2 45656 Lemma 2 for rhmsubc 45659. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
 
Theoremrhmsubclem3 45657* Lemma 3 for rhmsubc 45659. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
 
Theoremrhmsubclem4 45658* Lemma 4 for rhmsubc 45659. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
 
Theoremrhmsubc 45659 According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 ∈ (Subcat‘(RngCat‘𝑈)))
 
Theoremrhmsubccat 45660 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → ((RngCat‘𝑈) ↾cat 𝐻) ∈ Cat)
 
TheoremsrhmsubcALTVlem1 45661* Lemma 1 for srhmsubcALTV 45663. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈)))
 
TheoremsrhmsubcALTVlem2 45662* Lemma 2 for srhmsubcALTV 45663. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       ((𝑈𝑉 ∧ (𝑋𝐶𝑌𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCatALTV‘𝑈))𝑌))
 
TheoremsrhmsubcALTV 45663* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of special ring homomorphisms (i.e., ring homomorphisms from a special ring to another ring of that kind) is a subcategory of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
 
TheoremsringcatALTV 45664* The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)
 
TheoremcrhmsubcALTV 45665* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
 
TheoremcringcatALTV 45666* The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ CRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)
 
TheoremdrhmsubcALTV 45667* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of division ring homomorphisms is a "subcategory" of the category of (unital) rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐽 ∈ (Subcat‘(RingCatALTV‘𝑈)))
 
TheoremdrngcatALTV 45668* The restriction of the category of (unital) rings to the set of division ring homomorphisms is a category, the "category of division rings". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐽) ∈ Cat)
 
TheoremfldcatALTV 45669* The restriction of the category of (unital) rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → ((RingCatALTV‘𝑈) ↾cat 𝐹) ∈ Cat)
 
TheoremfldcALTV 45670* The restriction of the category of division rings to the set of field homomorphisms is a category, the "category of fields". (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉 → (((RingCatALTV‘𝑈) ↾cat 𝐽) ↾cat 𝐹) ∈ Cat)
 
TheoremfldhmsubcALTV 45671* According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of field homomorphisms is a "subcategory" of the category of division rings. (Contributed by AV, 20-Feb-2020.) (New usage is discouraged.)
𝐶 = (𝑈 ∩ DivRing)    &   𝐽 = (𝑟𝐶, 𝑠𝐶 ↦ (𝑟 RingHom 𝑠))    &   𝐷 = (𝑈 ∩ Field)    &   𝐹 = (𝑟𝐷, 𝑠𝐷 ↦ (𝑟 RingHom 𝑠))       (𝑈𝑉𝐹 ∈ (Subcat‘((RingCatALTV‘𝑈) ↾cat 𝐽)))
 
TheoremrngcrescrhmALTV 45672 The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → (𝐶cat 𝐻) = ((𝐶s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))
 
TheoremrhmsubcALTVlem1 45673 Lemma 1 for rhmsubcALTV 45677. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))
 
TheoremrhmsubcALTVlem2 45674 Lemma 2 for rhmsubcALTV 45677. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
 
TheoremrhmsubcALTVlem3 45675* Lemma 3 for rhmsubcALTV 45677. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCatALTV‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
 
TheoremrhmsubcALTVlem4 45676* Lemma 4 for rhmsubcALTV 45677. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((((𝜑𝑥𝑅) ∧ (𝑦𝑅𝑧𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCatALTV‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
 
TheoremrhmsubcALTV 45677 According to df-subc 17533, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17564 and subcss2 17567). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 ∈ (Subcat‘(RngCatALTV‘𝑈)))
 
TheoremrhmsubcALTVcat 45678 The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.) (New usage is discouraged.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCatALTV‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑 → ((RngCatALTV‘𝑈) ↾cat 𝐻) ∈ Cat)
 
20.41.20  Basic algebraic structures (extension)
 
20.41.20.1  Auxiliary theorems
 
Theoremopeliun2xp 45679 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5655. (Contributed by AV, 30-Mar-2019.)
(⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))
 
Theoremeliunxp2 45680* Membership in a union of Cartesian products over its second component, analogous to eliunxp 5749. (Contributed by AV, 30-Mar-2019.)
(𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))
 
Theoremmpomptx2 45681* 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 7396. (Contributed by AV, 30-Mar-2019.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)
 
Theoremcbvmpox2 45682* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7378 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7377. (Contributed by AV, 30-Mar-2019.)
𝑧𝐴    &   𝑦𝐷    &   𝑧𝐶    &   𝑤𝐶    &   𝑥𝐸    &   𝑦𝐸    &   (𝑦 = 𝑧𝐴 = 𝐷)    &   ((𝑦 = 𝑧𝑥 = 𝑤) → 𝐶 = 𝐸)       (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤𝐷, 𝑧𝐵𝐸)
 
Theoremdmmpossx2 45683* 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 7915. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})
 
Theoremmpoexxg2 45684* 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 7925. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)
 
Theoremovmpordxf 45685* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7432. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)    &   𝑥𝜑    &   𝑦𝜑    &   𝑦𝐴    &   𝑥𝐵    &   𝑥𝑆    &   𝑦𝑆       (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpordx 45686* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7432. (Contributed by AV, 30-Mar-2019.)
(𝜑𝐹 = (𝑥𝐶, 𝑦𝐷𝑅))    &   ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅 = 𝑆)    &   ((𝜑𝑦 = 𝐵) → 𝐶 = 𝐿)    &   (𝜑𝐴𝐿)    &   (𝜑𝐵𝐷)    &   (𝜑𝑆𝑋)       (𝜑 → (𝐴𝐹𝐵) = 𝑆)
 
Theoremovmpox2 45687* The value of an operation class abstraction. Variant of ovmpoga 7436 which does not require 𝐷 and 𝑥 to be distinct. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
((𝑥 = 𝐴𝑦 = 𝐵) → 𝑅 = 𝑆)    &   (𝑦 = 𝐵𝐶 = 𝐿)    &   𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)       ((𝐴𝐿𝐵𝐷𝑆𝐻) → (𝐴𝐹𝐵) = 𝑆)
 
Theoremfdmdifeqresdif 45688* The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
𝐹 = (𝑥𝐷 ↦ if(𝑥 = 𝑌, 𝑋, (𝐺𝑥)))       (𝐺:(𝐷 ∖ {𝑌})⟶𝑅𝐺 = (𝐹 ↾ (𝐷 ∖ {𝑌})))
 
Theoremoffvalfv 45689* The function operation expressed as a mapping with function values. (Contributed by AV, 6-Apr-2019.)
(𝜑𝐴𝑉)    &   (𝜑𝐹 Fn 𝐴)    &   (𝜑𝐺 Fn 𝐴)       (𝜑 → (𝐹f 𝑅𝐺) = (𝑥𝐴 ↦ ((𝐹𝑥)𝑅(𝐺𝑥))))
 
Theoremofaddmndmap 45690 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)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑉𝑌 ∧ (𝐴 ∈ (𝑅m 𝑉) ∧ 𝐵 ∈ (𝑅m 𝑉))) → (𝐴f + 𝐵) ∈ (𝑅m 𝑉))
 
Theoremmapsnop 45691 A singleton of an ordered pair as an element of the mapping operation. (Contributed by AV, 12-Apr-2019.)
𝐹 = {⟨𝑋, 𝑌⟩}       ((𝑋𝑉𝑌𝑅𝑅𝑊) → 𝐹 ∈ (𝑅m {𝑋}))
 
Theoremfprmappr 45692 A function with a domain of two elements as element of the mapping operator applied to a pair. (Contributed by AV, 20-May-2024.)
((𝑋𝑉 ∧ (𝐴𝑈𝐵𝑊𝐴𝐵) ∧ (𝐶𝑋𝐷𝑋)) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ (𝑋m {𝐴, 𝐵}))
 
Theoremmapprop 45693 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⟩}       (((𝑋𝑉𝐴𝑅) ∧ (𝑌𝑉𝐵𝑅) ∧ (𝑋𝑌𝑅𝑊)) → 𝐹 ∈ (𝑅m {𝑋, 𝑌}))
 
Theoremztprmneprm 45694 A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵𝐴 = 𝐵))
 
Theorem2t6m3t4e0 45695 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
((2 · 6) − (3 · 4)) = 0
 
Theoremssnn0ssfz 45696* For any finite subset of 0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 31117. (Contributed by AV, 30-Sep-2019.)
(𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛))
 
Theoremnn0sumltlt 45697 If the sum of two nonnegative integers is less than a third integer, then one of the summands is already less than this third integer. (Contributed by AV, 19-Oct-2019.)
((𝑎 ∈ ℕ0𝑏 ∈ ℕ0𝑐 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑐𝑏 < 𝑐))
 
20.41.20.2  The binomial coefficient operation (extension)
 
Theorembcpascm1 45698 Pascal's rule for the binomial coefficient, generalized to all integers 𝐾, shifted down by 1. (Contributed by AV, 8-Sep-2019.)
((𝑁 ∈ ℕ ∧ 𝐾 ∈ ℤ) → (((𝑁 − 1)C𝐾) + ((𝑁 − 1)C(𝐾 − 1))) = (𝑁C𝐾))
 
Theoremaltgsumbc 45699* The sum of binomial coefficients for a fixed positive 𝑁 with alternating signs is zero. Notice that this is not valid for 𝑁 = 0 (since ((-1↑0) · (0C0)) = (1 · 1) = 1). For a proof using Pascal's rule (bcpascm1 45698) instead of the binomial theorem (binom 15551) , see altgsumbcALT 45700. (Contributed by AV, 13-Sep-2019.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)
 
TheoremaltgsumbcALT 45700* Alternate proof of altgsumbc 45699, using Pascal's rule (bcpascm1 45698) instead of the binomial theorem (binom 15551). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)
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