P. 497 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 49601-49700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	funcringcsetcALTV2 49601*	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 49602	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 49603*	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 49604	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 49605	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 49606*	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 49607	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 49608*	Lemma for ringccatALTV 49609. (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 49609	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 49610	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 49611	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 49612	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 49613	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 49614	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 49615*	Lemma 1 for funcringcsetcALTV 49624. (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 49616*	Lemma 2 for funcringcsetcALTV 49624. (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 49617*	Lemma 3 for funcringcsetcALTV 49624. (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 49618*	Lemma 4 for funcringcsetcALTV 49624. (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 49619*	Lemma 5 for funcringcsetcALTV 49624. (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 49620*	Lemma 6 for funcringcsetcALTV 49624. (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 49621*	Lemma 7 for funcringcsetcALTV 49624. (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 49622*	Lemma 8 for funcringcsetcALTV 49624. (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 49623*	Lemma 9 for funcringcsetcALTV 49624. (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 49624*	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	srhmsubcALTVlem1 49625*	Lemma 1 for srhmsubcALTV 49627. (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 49626*	Lemma 2 for srhmsubcALTV 49627. (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 49627*	According to df-subc 18016, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 18044 and subcss2 18047). 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 49628*	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 49629*	According to df-subc 18016, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 18044 and subcss2 18047). 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 49630*	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 49631*	According to df-subc 18016, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 18044 and subcss2 18047). 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 49632*	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 49633*	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 49634*	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 49635*	According to df-subc 18016, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 18044 and subcss2 18047). 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)))
21.52.19  Prime rings (and integral domains) This section contains the definitions and teorems of section "Prime rings and integral domains" in JF's mathbox, transformed into extensible structures.
Syntax	cprmrng 49636	Extend class notation with the class of prime rings.
class PrmRing
Definition	df-prmring 49637	Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.)
⊢ PrmRing = {𝑟 ∈ Ring ∣ {(0g‘𝑟)} ∈ (PrmIdeal‘𝑟)}
Theorem	isprmrng 49638	The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑃 = (PrmIdeal‘𝑅)    ⇒   ⊢ (𝑅 ∈ PrmRing ↔ (𝑅 ∈ Ring ∧ { 0 } ∈ 𝑃))
Theorem	prmringnzring 49639	A prime ring is a nonzero ring. (Contributed by AV, 26-Jun-2026.)
⊢ (𝑅 ∈ PrmRing → 𝑅 ∈ NzRing)
Theorem	prmrngring 49640	A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 18-Jun-2026.) (Proof shortened by AV, 26-Jun-2026.)
⊢ (𝑅 ∈ PrmRing → 𝑅 ∈ Ring)
Theorem	smprngprmrng 49641	A simple ring (a nonzero ring whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 18-Jun-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (LIdeal‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝑈 = {{ 0 }, 𝐵}) → 𝑅 ∈ PrmRing)
Theorem	drngprmrng 49642	A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 18-Jun-2026.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ PrmRing)
Theorem	crngprmringidom 49643	A commutative ring is a prime ring if and only if it is an integral domain. (Contributed by AV, 27-Jun-2026.)
⊢ (𝑅 ∈ CRing → (𝑅 ∈ PrmRing ↔ 𝑅 ∈ IDomn))
Theorem	crngprmringdom 49644	A commutative ring is a prime ring if and only if it is a domain. (Contributed by AV, 27-Jun-2026.)
⊢ (𝑅 ∈ CRing → (𝑅 ∈ PrmRing ↔ 𝑅 ∈ Domn))
Theorem	dfidom2 49645	Alternate definition of the class of integral domains. An integral domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
⊢ IDomn = (PrmRing ∩ CRing)
Theorem	isidom2 49646	The predicate "is an integral domain": An integral domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by AV, 27-Jun-2026.)
⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ PrmRing ∧ 𝑅 ∈ CRing))
Theorem	isidom3 49647*	The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 0 ≠ 1 ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 ))))
Theorem	idomnzd 49648	A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by AV, 12-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑋 · 𝑌) = 0 )) → (𝑋 = 0 ∨ 𝑌 = 0 ))
Theorem	idomcanl 49649	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 13-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 ≠ 0 ) → ((𝑋 · 𝑌) = (𝑋 · 𝑍) → 𝑌 = 𝑍))
Theorem	idomcanr 49650	Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.) (Revised by AV, 13-Jul-2026.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ IDomn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑍 ≠ 0 ) → ((𝑋 · 𝑍) = (𝑌 · 𝑍) → 𝑋 = 𝑌))
21.52.20  Basic algebraic structures (extension)
21.52.20.1  Auxiliary theorems
Theorem	eliunxp2 49651*	Membership in a union of Cartesian products over its second component, analogous to eliunxp 5820. (Contributed by AV, 30-Mar-2019.)
⊢ (𝐶 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↔ ∃𝑥∃𝑦(𝐶 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
Theorem	mpomptx2 49652*	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 7556. (Contributed by AV, 30-Mar-2019.)
⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)    ⇒   ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦}) ↦ 𝐶) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐷)
Theorem	cbvmpox2 49653*	Rule to change the bound variable in a maps-to function, using implicit substitution. This version of cbvmpo 7537 allows 𝐴 to be a function of 𝑦, analogous to cbvmpox 7536. (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 49654*	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 8102. (Contributed by AV, 30-Mar-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ dom 𝐹 ⊆ ∪ 𝑦 ∈ 𝐵 (𝐴 × {𝑦})
Theorem	mpoexxg2 49655*	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 8113. (Contributed by AV, 30-Mar-2019.)
⊢ 𝐹 = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)    ⇒   ⊢ ((𝐵 ∈ 𝑅 ∧ ∀𝑦 ∈ 𝐵 𝐴 ∈ 𝑆) → 𝐹 ∈ V)
Theorem	ovmpordxf 49656*	Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7594. (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 49657*	Value of an operation given by a maps-to rule, deduction form, with substitution of second argument, analogous to ovmpodxf 7594. (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 49658*	The value of an operation class abstraction. Variant of ovmpoga 7598 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 49659*	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	ofaddmndmap 49660	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 49661	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 49662	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 49663	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}))
Theorem	ztprmneprm 49664	A prime is not an integer multiple of another prime. (Contributed by AV, 23-May-2019.)
⊢ ((𝑍 ∈ ℤ ∧ 𝐴 ∈ ℙ ∧ 𝐵 ∈ ℙ) → ((𝑍 · 𝐴) = 𝐵 → 𝐴 = 𝐵))
Theorem	2t6m3t4e0 49665	2 times 6 minus 3 times 4 equals 0. (Contributed by AV, 24-May-2019.)
⊢ ((2 · 6) − (3 · 4)) = 0
Theorem	ssnn0ssfz 49666*	For any finite subset of ℕ0, find a superset in the form of a set of sequential integers, analogous to ssnnssfz 33449. (Contributed by AV, 30-Sep-2019.)
⊢ (𝐴 ∈ (𝒫 ℕ0 ∩ Fin) → ∃𝑛 ∈ ℕ0 𝐴 ⊆ (0...𝑛))
Theorem	nn0sumltlt 49667	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 ∈ ℕ0 ∧ 𝑏 ∈ ℕ0 ∧ 𝑐 ∈ ℕ0) → ((𝑎 + 𝑏) < 𝑐 → 𝑏 < 𝑐))
21.52.20.2  The binomial coefficient operation (extension)
Theorem	bcpascm1 49668	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))
Theorem	altgsumbc 49669*	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 49668) instead of the binomial theorem (binom 16028), see altgsumbcALT 49670. (Contributed by AV, 13-Sep-2019.)
⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)(( -1↑𝑘) · (𝑁C𝑘)) = 0)
Theorem	altgsumbcALT 49670*	Alternate proof of altgsumbc 49669, using Pascal's rule (bcpascm1 49668) instead of the binomial theorem (binom 16028). (Contributed by AV, 8-Sep-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑁 ∈ ℕ → Σ𝑘 ∈ (0...𝑁)(( -1↑𝑘) · (𝑁C𝑘)) = 0)
21.52.20.3  The ` ZZ `-module ` ZZ X. ZZ `
Theorem	zlmodzxzlmod 49671	The ℤ-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))
Theorem	zlmodzxzel 49672	An element of the (base set of the) ℤ-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))
Theorem	zlmodzxz0 49673	The 0 of the ℤ-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 = (0g‘𝑍)
Theorem	zlmodzxzscm 49674	The scalar multiplication of the ℤ-module ℤ × ℤ. (Contributed by AV, 20-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ ∙ = ( ·𝑠 ‘𝑍)    ⇒   ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐴 ∙ {⟨0, 𝐵⟩, ⟨1, 𝐶⟩}) = {⟨0, (𝐴 · 𝐵)⟩, ⟨1, (𝐴 · 𝐶)⟩})
Theorem	zlmodzxzadd 49675	The addition of the ℤ-module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ + = (+g‘𝑍)    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} + {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 + 𝐵)⟩, ⟨1, (𝐶 + 𝐷)⟩})
Theorem	zlmodzxzsubm 49676	The subtraction of the ℤ-module ℤ × ℤ expressed as addition. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ − = (-g‘𝑍)    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} − {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} (+g‘𝑍)( -1( ·𝑠 ‘𝑍){⟨0, 𝐵⟩, ⟨1, 𝐷⟩})))
Theorem	zlmodzxzsub 49677	The subtraction of the ℤ-module ℤ × ℤ. (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
⊢ 𝑍 = (ℤring freeLMod {0, 1})    &   ⊢ − = (-g‘𝑍)    ⇒   ⊢ (((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝐶 ∈ ℤ ∧ 𝐷 ∈ ℤ)) → ({⟨0, 𝐴⟩, ⟨1, 𝐶⟩} − {⟨0, 𝐵⟩, ⟨1, 𝐷⟩}) = {⟨0, (𝐴 − 𝐵)⟩, ⟨1, (𝐶 − 𝐷)⟩})
21.52.20.4  Group sum operation (extension 2)
Theorem	mgpsumunsn 49678*	Extract a summand/factor from the group sum for the multiplicative group of a unital ring. (Contributed by AV, 29-Dec-2018.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ⊢ (𝜑 → 𝑋 ∈ (Base‘𝑅))    &   ⊢ (𝑘 = 𝐼 → 𝐴 = 𝑋)    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = ((𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)) · 𝑋))
Theorem	mgpsumz 49679*	If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the zero of the ring, the group sum itself is zero. (Contributed by AV, 29-Dec-2018.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝑘 = 𝐼 → 𝐴 = 0 )    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = 0 )
Theorem	mgpsumn 49680*	If the group sum for the multiplicative group of a unital ring contains a summand/factor that is the one of the ring, this summand/ factor can be removed from the group sum. (Contributed by AV, 29-Dec-2018.)
⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ CRing)    &   ⊢ (𝜑 → 𝑁 ∈ Fin)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑁)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑁) → 𝐴 ∈ (Base‘𝑅))    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ (𝑘 = 𝐼 → 𝐴 = 1 )    ⇒   ⊢ (𝜑 → (𝑀 Σg (𝑘 ∈ 𝑁 ↦ 𝐴)) = (𝑀 Σg (𝑘 ∈ (𝑁 ∖ {𝐼}) ↦ 𝐴)))
21.52.20.5  Symmetric groups (extension)
Theorem	exple2lt6 49681	A nonnegative integer to the power of itself is less than 6 if it is less than or equal to 2. (Contributed by AV, 16-Mar-2019.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 2) → (𝑁↑𝑁) < 6)
Theorem	pgrple2abl 49682	Every symmetric group on a set with at most 2 elements is abelian. (Contributed by AV, 16-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ (♯‘𝐴) ≤ 2) → 𝐺 ∈ Abel)
Theorem	pgrpgt2nabl 49683	Every symmetric group on a set with more than 2 elements is not abelian, see also the remark in [Rotman] p. 28. (Contributed by AV, 21-Mar-2019.)
⊢ 𝐺 = (SymGrp‘𝐴)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 2 < (♯‘𝐴)) → 𝐺 ∉ Abel)
21.52.20.6  Divisibility (extension)
Theorem	invginvrid 49684	Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑁 = (invg‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋)) = 𝑋)
21.52.20.7  The support of functions (extension)
Theorem	rmsupp0 49685*	The support of a mapping of a multiplication of zero with a function into a ring is empty. (Contributed by AV, 10-Apr-2019.)
⊢ 𝑅 = (Base‘𝑀)    ⇒   ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 = (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = ∅)
Theorem	domnmsuppn0 49686*	The support of a mapping of a multiplication of a nonzero constant with a function into a (ring theoretic) domain equals the support of the function. (Contributed by AV, 11-Apr-2019.)
⊢ 𝑅 = (Base‘𝑀)    ⇒   ⊢ (((𝑀 ∈ Domn ∧ 𝑉 ∈ 𝑋) ∧ (𝐶 ∈ 𝑅 ∧ 𝐶 ≠ (0g‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) = (𝐴 supp (0g‘𝑀)))
Theorem	rmsuppss 49687*	The support of a mapping of a multiplication of a constant with a function into a ring is a subset of the support of the function. (Contributed by AV, 11-Apr-2019.)
⊢ 𝑅 = (Base‘𝑀)    ⇒   ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑀)))
Theorem	scmsuppss 49688*	The support of a mapping of a scalar multiplication with a function of scalars is a subset of the support of the function of scalars. (Contributed by AV, 5-Apr-2019.)
⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉)) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ⊆ (𝐴 supp (0g‘𝑆)))
21.52.20.8  Finitely supported functions (extension)
Theorem	rmsuppfi 49689*	The support of a mapping of a multiplication of a constant with a function into a ring is finite if the support of the function is finite. (Contributed by AV, 11-Apr-2019.)
⊢ 𝑅 = (Base‘𝑀)    ⇒   ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑀)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) supp (0g‘𝑀)) ∈ Fin)
Theorem	rmfsupp 49690*	A mapping of a multiplication of a constant with a function into a ring is finitely supported if the function is finitely supported. (Contributed by AV, 9-Jun-2019.)
⊢ 𝑅 = (Base‘𝑀)    ⇒   ⊢ (((𝑀 ∈ Ring ∧ 𝑉 ∈ 𝑋 ∧ 𝐶 ∈ 𝑅) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑀)) → (𝑣 ∈ 𝑉 ↦ (𝐶(.r‘𝑀)(𝐴‘𝑣))) finSupp (0g‘𝑀))
Theorem	scmsuppfi 49691*	The support of a mapping of a scalar multiplication with a function of scalars is finite if the support of the function of scalars is finite. (Contributed by AV, 5-Apr-2019.)
⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ (𝐴 supp (0g‘𝑆)) ∈ Fin) → ((𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) supp (0g‘𝑀)) ∈ Fin)
Theorem	scmfsupp 49692*	A mapping of a scalar multiplication with a function of scalars is finitely supported if the function of scalars is finitely supported. (Contributed by AV, 9-Jun-2019.)
⊢ 𝑆 = (Scalar‘𝑀)    &   ⊢ 𝑅 = (Base‘𝑆)    ⇒   ⊢ (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐴 ∈ (𝑅 ↑m 𝑉) ∧ 𝐴 finSupp (0g‘𝑆)) → (𝑣 ∈ 𝑉 ↦ ((𝐴‘𝑣)( ·𝑠 ‘𝑀)𝑣)) finSupp (0g‘𝑀))
Theorem	suppmptcfin 49693*	The support of a mapping with value 0 except of one is finite. (Contributed by AV, 27-Apr-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 supp 0 ) ∈ Fin)
Theorem	mptcfsupp 49694*	A mapping with value 0 except of one is finitely supported. (Contributed by AV, 9-Jun-2019.)
⊢ 𝐵 = (Base‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵 ∧ 𝑋 ∈ 𝑉) → 𝐹 finSupp 0 )
Theorem	fsuppmptdmf 49695*	A mapping with a finite domain is finitely supported. (Contributed by AV, 4-Sep-2019.)
⊢ Ⅎ𝑥𝜑    &   ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝑌)    &   ⊢ (𝜑 → 𝐴 ∈ Fin)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝐹 finSupp 𝑍)
21.52.20.9  Left modules (extension)
Theorem	lmodvsmdi 49696	Multiple distributive law for scalar product (left-distributivity). (Contributed by AV, 5-Sep-2019.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ ↑ = (.g‘𝑊)    &   ⊢ 𝐸 = (.g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ LMod ∧ (𝑅 ∈ 𝐾 ∧ 𝑁 ∈ ℕ0 ∧ 𝑋 ∈ 𝑉)) → (𝑅 · (𝑁 ↑ 𝑋)) = ((𝑁𝐸𝑅) · 𝑋))
Theorem	gsumlsscl 49697*	Closure of a group sum in a linear subspace: A (finitely supported) sum of scalar multiplications of vectors of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
⊢ 𝑆 = (LSubSp‘𝑀)    &   ⊢ 𝑅 = (Scalar‘𝑀)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑀 ∈ LMod ∧ 𝑍 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑍) → ((𝐹 ∈ (𝐵 ↑m 𝑉) ∧ 𝐹 finSupp (0g‘𝑅)) → (𝑀 Σg (𝑣 ∈ 𝑉 ↦ ((𝐹‘𝑣)( ·𝑠 ‘𝑀)𝑣))) ∈ 𝑍))
21.52.20.10  Associative algebras (extension)
Theorem	assaascl0 49698	The scalar 0 embedded into an associative algebra corresponds to the 0 of the associative algebra. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → (𝐴‘(0g‘𝐹)) = (0g‘𝑊))
Theorem	assaascl1 49699	The scalar 1 embedded into an associative algebra corresponds to the 1 of the an associative algebra. (Contributed by AV, 31-Jul-2019.)
⊢ 𝐴 = (algSc‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ (𝜑 → 𝑊 ∈ AssAlg)    ⇒   ⊢ (𝜑 → (𝐴‘(1r‘𝐹)) = (1r‘𝑊))
21.52.20.11  Univariate polynomials (extension)
Theorem	ply1vr1smo 49700	The variable in a polynomial expressed as scaled monomial. (Contributed by AV, 12-Aug-2019.)
⊢ 𝑃 = (Poly1‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝑃)    &   ⊢ 𝐺 = (mulGrp‘𝑃)    &   ⊢ ↑ = (.g‘𝐺)    &   ⊢ 𝑋 = (var1‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → ( 1 · (1 ↑ 𝑋)) = 𝑋)