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Theorem List for Metamath Proof Explorer - 43801-43900   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremfuncringcsetcALTV2lem3 43801* Lemma 3 for funcringcsetcALTV2 43808. (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 43802* Lemma 4 for funcringcsetcALTV2 43808. (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 43803* Lemma 5 for funcringcsetcALTV2 43808. (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 43804* Lemma 6 for funcringcsetcALTV2 43808. (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 43805* Lemma 7 for funcringcsetcALTV2 43808. (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 43806* Lemma 8 for funcringcsetcALTV2 43808. (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 43807* Lemma 9 for funcringcsetcALTV2 43808. (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 43808* 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 43809 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 43810* 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 43811 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 43812 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 43813* 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 43814 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 43815* Lemma for ringccatALTV 43816. (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 43816 The category of rings is a category. (Contributed by AV, 14-Feb-2020.) (New usage is discouraged.)
𝐶 = (RingCatALTV‘𝑈)       (𝑈𝑉𝐶 ∈ Cat)

TheoremringcidALTV 43817 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 43818 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 43819 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 43820 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 43821 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 43822* Lemma 1 for funcringcsetcALTV 43831. (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 43823* Lemma 2 for funcringcsetcALTV 43831. (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 43824* Lemma 3 for funcringcsetcALTV 43831. (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 43825* Lemma 4 for funcringcsetcALTV 43831. (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 43826* Lemma 5 for funcringcsetcALTV 43831. (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 43827* Lemma 6 for funcringcsetcALTV 43831. (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 43828* Lemma 7 for funcringcsetcALTV 43831. (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 43829* Lemma 8 for funcringcsetcALTV 43831. (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 43830* Lemma 9 for funcringcsetcALTV 43831. (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 43831* 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 43832 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 43833 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 43834* 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 43835 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) = ∅)

20.38.18.10  Subcategories of the category of rings

Theoremsrhmsubclem1 43836* Lemma 1 for srhmsubc 43839. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       (𝑋𝐶𝑋 ∈ (𝑈 ∩ Ring))

Theoremsrhmsubclem2 43837* Lemma 2 for srhmsubc 43839. (Contributed by AV, 19-Feb-2020.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈)))

Theoremsrhmsubclem3 43838* Lemma 3 for srhmsubc 43839. (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 43839* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43840* 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 43841* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43842* 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 43843* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43844* 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 43845* 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 43846* 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 43847* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43848 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 43849 Lemma 1 for rhmsubc 43853. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       (𝜑𝐻 Fn (𝑅 × 𝑅))

Theoremrhmsubclem2 43850 Lemma 2 for rhmsubc 43853. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑋𝑅𝑌𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))

Theoremrhmsubclem3 43851* Lemma 3 for rhmsubc 43853. (Contributed by AV, 2-Mar-2020.)
(𝜑𝑈𝑉)    &   𝐶 = (RngCat‘𝑈)    &   (𝜑𝑅 = (Ring ∩ 𝑈))    &   𝐻 = ( RingHom ↾ (𝑅 × 𝑅))       ((𝜑𝑥𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))

Theoremrhmsubclem4 43852* Lemma 4 for rhmsubc 43853. (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 43853 According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43854 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 43855* Lemma 1 for srhmsubcALTV 43857. (Contributed by AV, 19-Feb-2020.) (New usage is discouraged.)
𝑟𝑆 𝑟 ∈ Ring    &   𝐶 = (𝑈𝑆)       ((𝑈𝑉𝑋𝐶) → 𝑋 ∈ (Base‘(RingCatALTV‘𝑈)))

TheoremsrhmsubcALTVlem2 43856* Lemma 2 for srhmsubcALTV 43857. (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 43857* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43858* 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 43859* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43860* 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 43861* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43862* 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 43863* 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 43864* 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 43865* According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43866 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 43867 Lemma 1 for rhmsubcALTV 43871. (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 43868 Lemma 2 for rhmsubcALTV 43871. (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 43869* Lemma 3 for rhmsubcALTV 43871. (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 43870* Lemma 4 for rhmsubcALTV 43871. (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 43871 According to df-subc 16911, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 16939 and subcss2 16942). 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 43872 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)

20.38.19  Basic algebraic structures (extension)

20.38.19.1  Auxiliary theorems

Theoremopeliun2xp 43873 Membership of an ordered pair in a union of Cartesian products over its second component, analogous to opeliunxp 5508. (Contributed by AV, 30-Mar-2019.)
(⟨𝐶, 𝑦⟩ ∈ 𝑦𝐵 (𝐴 × {𝑦}) ↔ (𝑦𝐵𝐶𝐴))

Theoremeliunxp2 43874* Membership in a union of Cartesian products over its second component, analogous to eliunxp 5597. (Contributed by AV, 30-Mar-2019.)
(𝐶 𝑦𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥𝐴𝑦𝐵)))

Theoremmpomptx2 43875* 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 7124. (Contributed by AV, 30-Mar-2019.)
(𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑦𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Theoremcbvmpox2 43876* Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7107 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7106. (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 43877* 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 7623. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       dom 𝐹 𝑦𝐵 (𝐴 × {𝑦})

Theoremmpoexxg2 43878* 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 7632. (Contributed by AV, 30-Mar-2019.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       ((𝐵𝑅 ∧ ∀𝑦𝐵 𝐴𝑆) → 𝐹 ∈ V)

Theoremovmpordxf 43879* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7159. (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 43880* Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7159. (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 43881* The value of an operation class abstraction. Variant of ovmpoga 7163 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 43882* 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 43883* 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 43884 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 43885 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}))

Theoremmapprop 43886 An unordered pair containing two ordered pairs as an element of the mapping operation. (Contributed by AV, 16-Apr-2019.)
𝐹 = {⟨𝑋, 𝐴⟩, ⟨𝑌, 𝐵⟩}       (((𝑋𝑉𝐴𝑅) ∧ (𝑌𝑉𝐵𝑅) ∧ (𝑋𝑌𝑅𝑊)) → 𝐹 ∈ (𝑅𝑚 {𝑋, 𝑌}))

Theoremztprmneprm 43887 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 43888 2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
((2 · 6) − (3 · 4)) = 0

Theoremssnn0ssfz 43889* For any finite subset of 0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 30190. (Contributed by AV, 30-Sep-2019.)
(𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛))

Theoremnn0sumltlt 43890 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))

20.38.19.2  The binomial coefficient operation (extension)

Theorembcpascm1 43891 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 43892* 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 43891) instead of the binomial theorem (binom 15018) , see altgsumbcALT 43893. (Contributed by AV, 13-Sep-2019.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)

TheoremaltgsumbcALT 43893* Alternate proof of altgsumbc 43892, using Pascal's rule (bcpascm1 43891) instead of the binomial theorem (binom 15018). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)((-1↑𝑘) · (𝑁C𝑘)) = 0)

20.38.19.3  The ` ZZ `-module ` ZZ X. ZZ `

Theoremzlmodzxzlmod 43894 The -module ℤ × ℤ is a (left) module with the ring of integers as base set. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})       (𝑍 ∈ LMod ∧ ℤring = (Scalar‘𝑍))

Theoremzlmodzxzel 43895 An element of the (base set of the) -module ℤ × ℤ. (Contributed by AV, 21-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ (Base‘𝑍))

Theoremzlmodzxz0 43896 The 0 of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}        0 = (0g𝑍)

Theoremzlmodzxzscm 43897 The scalar multiplication of the -module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = ( ·𝑠𝑍)       ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 {⟨0, 𝐵⟩, ⟨1, 𝐶⟩}) = {⟨0, (𝐴 · 𝐵)⟩, ⟨1, (𝐴 · 𝐶)⟩})

Theoremzlmodzxzadd 43898 The addition of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    + = (+g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} + {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩})

Theoremzlmodzxzsubm 43899 The subtraction of the -module ℤ × ℤ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} (+g𝑍)(-1( ·𝑠𝑍){⟨0, 𝐵⟩, ⟨1, 𝐷⟩})))

Theoremzlmodzxzsub 43900 The subtraction of the -module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    = (-g𝑍)       (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴𝐵)⟩, ⟨1, (𝐶𝐷)⟩})

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