P. 207 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20601-20700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ringcco 20601	Composition in the category of unital rings. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 8-Mar-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐹:(Base‘𝑋)⟶(Base‘𝑌))    &   ⊢ (𝜑 → 𝐺:(Base‘𝑌)⟶(Base‘𝑍))    ⇒   ⊢ (𝜑 → (𝐺(⟨𝑋, 𝑌⟩ · 𝑍)𝐹) = (𝐺 ∘ 𝐹))
Theorem	dfringc2 20602	Alternate definition of the category of unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    &   ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	rhmsscmap2 20603*	The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    ⇒   ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Theorem	rhmsscmap 20604*	The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the mappings between base sets of extensible structures (in the same universe). (Contributed by AV, 9-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    ⇒   ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Theorem	rhmsubcsetclem1 20605	Lemma 1 for rhmsubcsetc 20607. (Contributed by AV, 9-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
Theorem	rhmsubcsetclem2 20606*	Lemma 2 for rhmsubcsetc 20607. (Contributed by AV, 9-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
Theorem	rhmsubcsetc 20607	The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of extensible structures. (Contributed by AV, 9-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))
Theorem	ringccat 20608	The category of unital rings is a category. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 9-Mar-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐶 ∈ Cat)
Theorem	ringcid 20609	The identity arrow in the category of unital rings is the identity function. (Contributed by AV, 14-Feb-2020.) (Revised by AV, 10-Mar-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ 1 = (Id‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ 𝑆 = (Base‘𝑋)    ⇒   ⊢ (𝜑 → ( 1 ‘𝑋) = ( I ↾ 𝑆))
Theorem	rhmsscrnghm 20610	The unital ring homomorphisms between unital rings (in a universe) are a subcategory subset of the non-unital ring homomorphisms between non-unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝑆 = (Rng ∩ 𝑈))    ⇒   ⊢ (𝜑 → ( RingHom ↾ (𝑅 × 𝑅)) ⊆cat ( RngHom ↾ (𝑆 × 𝑆)))
Theorem	rhmsubcrngclem1 20611	Lemma 1 for rhmsubcrngc 20613. (Contributed by AV, 9-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
Theorem	rhmsubcrngclem2 20612*	Lemma 2 for rhmsubcrngc 20613. (Contributed by AV, 12-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))
Theorem	rhmsubcrngc 20613	The unital ring homomorphisms between unital rings (in a universe) are a subcategory of the category of non-unital rings. (Contributed by AV, 12-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ (𝜑 → 𝐻 ∈ (Subcat‘𝐶))
Theorem	rngcresringcat 20614	The restriction of the category of non-unital rings to the set of unital ring homomorphisms is the category of unital rings. (Contributed by AV, 16-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Ring ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = (RingCat‘𝑈))
Theorem	ringcsect 20615	A section in the category of unital rings, written out. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐸 = (Base‘𝑋)    &   ⊢ 𝑆 = (Sect‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑆𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingHom 𝑌) ∧ 𝐺 ∈ (𝑌 RingHom 𝑋) ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐸))))
Theorem	ringcinv 20616	An inverse in the category of unital rings is the converse operation. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝑁 = (Inv‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹(𝑋𝑁𝑌)𝐺 ↔ (𝐹 ∈ (𝑋 RingIso 𝑌) ∧ 𝐺 = ◡𝐹)))
Theorem	ringciso 20617	An isomorphism in the category of unital rings is a bijection. (Contributed by AV, 14-Feb-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ 𝐼 = (Iso‘𝐶)    ⇒   ⊢ (𝜑 → (𝐹 ∈ (𝑋𝐼𝑌) ↔ 𝐹 ∈ (𝑋 RingIso 𝑌)))
Theorem	ringcbasbas 20618	An element of the base set of the base set of the category of unital rings (i.e. the base set of a ring) belongs to the considered weak universe. (Contributed by AV, 15-Feb-2020.)
⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    ⇒   ⊢ ((𝜑 ∧ 𝑅 ∈ 𝐵) → (Base‘𝑅) ∈ 𝑈)
Theorem	funcringcsetc 20619*	The "natural forgetful functor" from the category of unital rings into the category of sets which sends each ring to its underlying set (base set) and the morphisms (ring homomorphisms) to mappings of the corresponding base sets. (Contributed by AV, 26-Mar-2020.)
⊢ 𝑅 = (RingCat‘𝑈)    &   ⊢ 𝑆 = (SetCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑈 ∈ WUni)    &   ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐵 ↦ (Base‘𝑥)))    &   ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ ( I ↾ (𝑥 RingHom 𝑦))))    ⇒   ⊢ (𝜑 → 𝐹(𝑅 Func 𝑆)𝐺)
Theorem	zrtermoringc 20620	The zero ring is a terminal object in the category of unital rings. (Contributed by AV, 17-Apr-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    ⇒   ⊢ (𝜑 → 𝑍 ∈ (TermO‘𝐶))
Theorem	zrninitoringc 20621*	The zero ring is not an initial object in the category of unital rings (if the universe contains at least one unital ring different from the zero ring). (Contributed by AV, 18-Apr-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RingCat‘𝑈)    &   ⊢ (𝜑 → 𝑍 ∈ (Ring ∖ NzRing))    &   ⊢ (𝜑 → 𝑍 ∈ 𝑈)    &   ⊢ (𝜑 → ∃𝑟 ∈ (Base‘𝐶)𝑟 ∈ NzRing)    ⇒   ⊢ (𝜑 → 𝑍 ∉ (InitO‘𝐶))
10.3.14.3  Subcategories of the category of rings
Theorem	srhmsubclem1 20622*	Lemma 1 for srhmsubc 20625. (Contributed by AV, 19-Feb-2020.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    ⇒   ⊢ (𝑋 ∈ 𝐶 → 𝑋 ∈ (𝑈 ∩ Ring))
Theorem	srhmsubclem2 20623*	Lemma 2 for srhmsubc 20625. (Contributed by AV, 19-Feb-2020.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    ⇒   ⊢ ((𝑈 ∈ 𝑉 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ (Base‘(RingCat‘𝑈)))
Theorem	srhmsubclem3 20624*	Lemma 3 for srhmsubc 20625. (Contributed by AV, 19-Feb-2020.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ ((𝑈 ∈ 𝑉 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐽𝑌) = (𝑋(Hom ‘(RingCat‘𝑈))𝑌))
Theorem	srhmsubc 20625*	According to df-subc 17748, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17776 and subcss2 17779). Therefore, the set of special ring homomorphisms (i.e., ring homomorphisms from a special ring to another ring of that kind) is a subcategory of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
Theorem	sringcat 20626*	The restriction of the category of (unital) rings to the set of special ring homomorphisms is a category. (Contributed by AV, 19-Feb-2020.)
⊢ ∀𝑟 ∈ 𝑆 𝑟 ∈ Ring    &   ⊢ 𝐶 = (𝑈 ∩ 𝑆)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)
Theorem	crhmsubc 20627*	According to df-subc 17748, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17776 and subcss2 17779). Therefore, the set of commutative ring homomorphisms (i.e. ring homomorphisms from a commutative ring to a commutative ring) is a "subcategory" of the category of (unital) rings. (Contributed by AV, 19-Feb-2020.)
⊢ 𝐶 = (𝑈 ∩ CRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → 𝐽 ∈ (Subcat‘(RingCat‘𝑈)))
Theorem	cringcat 20628*	The restriction of the category of (unital) rings to the set of commutative ring homomorphisms is a category, the "category of commutative rings". (Contributed by AV, 19-Feb-2020.)
⊢ 𝐶 = (𝑈 ∩ CRing)    &   ⊢ 𝐽 = (𝑟 ∈ 𝐶, 𝑠 ∈ 𝐶 ↦ (𝑟 RingHom 𝑠))    ⇒   ⊢ (𝑈 ∈ 𝑉 → ((RingCat‘𝑈) ↾cat 𝐽) ∈ Cat)
Theorem	rngcrescrhm 20629	The category of non-unital rings (in a universe) restricted to the ring homomorphisms between unital rings (in the same universe). (Contributed by AV, 1-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → (𝐶 ↾cat 𝐻) = ((𝐶 ↾s 𝑅) sSet ⟨(Hom ‘ndx), 𝐻⟩))
Theorem	rhmsubclem1 20630	Lemma 1 for rhmsubc 20634. (Contributed by AV, 2-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → 𝐻 Fn (𝑅 × 𝑅))
Theorem	rhmsubclem2 20631	Lemma 2 for rhmsubc 20634. (Contributed by AV, 2-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑅 ∧ 𝑌 ∈ 𝑅) → (𝑋𝐻𝑌) = (𝑋 RingHom 𝑌))
Theorem	rhmsubclem3 20632*	Lemma 3 for rhmsubc 20634. (Contributed by AV, 2-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑅) → ((Id‘(RngCat‘𝑈))‘𝑥) ∈ (𝑥𝐻𝑥))
Theorem	rhmsubclem4 20633*	Lemma 4 for rhmsubc 20634. (Contributed by AV, 2-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑅) ∧ (𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑅)) ∧ (𝑓 ∈ (𝑥𝐻𝑦) ∧ 𝑔 ∈ (𝑦𝐻𝑧))) → (𝑔(⟨𝑥, 𝑦⟩(comp‘(RngCat‘𝑈))𝑧)𝑓) ∈ (𝑥𝐻𝑧))
Theorem	rhmsubc 20634	According to df-subc 17748, the subcategories (Subcat‘𝐶) of a category 𝐶 are subsets of the homomorphisms of 𝐶 (see subcssc 17776 and subcss2 17779). Therefore, the set of unital ring homomorphisms is a "subcategory" of the category of non-unital rings. (Contributed by AV, 2-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → 𝐻 ∈ (Subcat‘(RngCat‘𝑈)))
Theorem	rhmsubccat 20635	The restriction of the category of non-unital rings to the set of unital ring homomorphisms is a category. (Contributed by AV, 4-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑅 = (Ring ∩ 𝑈))    &   ⊢ 𝐻 = ( RingHom ↾ (𝑅 × 𝑅))    ⇒   ⊢ (𝜑 → ((RngCat‘𝑈) ↾cat 𝐻) ∈ Cat)
10.3.15  Left regular elements and domains
Syntax	crlreg 20636	Set of left-regular elements in a ring.
class RLReg
Syntax	cdomn 20637	Class of (ring theoretic) domains.
class Domn
Syntax	cidom 20638	Class of integral domains.
class IDomn
Definition	df-rlreg 20639*	Define the set of left-regular elements in a ring as those elements which are not left zero divisors, meaning that multiplying a nonzero element on the left by a left-regular element gives a nonzero product. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
Definition	df-domn 20640*	A domain is a nonzero ring in which there are no nontrivial zero divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ Domn = {𝑟 ∈ NzRing ∣ [(Base‘𝑟) / 𝑏][(0g‘𝑟) / 𝑧]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ((𝑥(.r‘𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧 ∨ 𝑦 = 𝑧))}
Definition	df-idom 20641	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ IDomn = (CRing ∩ Domn)
Theorem	rrgval 20642*	Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}
Theorem	isrrg 20643*	Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))
Theorem	rrgeq0i 20644	Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 → 𝑌 = 0 ))
Theorem	rrgeq0 20645	Left-multiplication by a left regular element does not change zeroness. (Contributed by Stefan O'Rear, 28-Mar-2015.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐸 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ 𝑌 = 0 ))
Theorem	rrgsupp 20646	Left multiplication by a left regular element does not change the support set of a vector. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Revised by AV, 20-Jul-2019.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐸)    &   ⊢ (𝜑 → 𝑌:𝐼⟶𝐵)    ⇒   ⊢ (𝜑 → (((𝐼 × {𝑋}) ∘f · 𝑌) supp 0 ) = (𝑌 supp 0 ))
Theorem	rrgss 20647	Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ 𝐸 ⊆ 𝐵
Theorem	unitrrg 20648	Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ 𝐸)
Theorem	rrgnz 20649	In a nonzero ring, the zero is a left zero divisor (that is, not a left-regular element). (Contributed by Thierry Arnoux, 6-May-2025.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → ¬ 0 ∈ 𝐸)
Theorem	isdomn 20650*	Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0 ∨ 𝑦 = 0 ))))
Theorem	domnnzr 20651	A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
Theorem	domnring 20652	A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
Theorem	domneq0 20653	In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 )))
Theorem	domnmuln0 20654	In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 )
Theorem	isdomn5 20655*	The equivalence between the right conjuncts in the right hand sides of isdomn 20650 and isdomn2 20656, in predicate calculus form. (Contributed by SN, 16-Sep-2024.)
⊢ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → (𝑎 = 0 ∨ 𝑏 = 0 )) ↔ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ((𝑎 · 𝑏) = 0 → 𝑏 = 0 ))
Theorem	isdomn2 20656	A ring is a domain iff all nonzero elements are regular elements. (Contributed by Mario Carneiro, 28-Mar-2015.) (Proof shortened by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
Theorem	isdomn2OLD 20657	Obsolete version of isdomn2 20656 as of 21-Jun-2025. (Contributed by Mario Carneiro, 28-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
Theorem	domnrrg 20658	In a domain, a nonzero element is a regular element. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → 𝑋 ∈ 𝐸)
Theorem	isdomn6 20659	A ring is a domain iff the regular elements are the nonzero elements. Compare isdomn2 20656, domnrrg 20658. (Contributed by Thierry Arnoux, 6-May-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) = 𝐸))
Theorem	isdomn3 20660	Nonzero elements form a multiplicative submonoid of any domain. (Contributed by Stefan O'Rear, 11-Sep-2015.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑈 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ Ring ∧ (𝐵 ∖ { 0 }) ∈ (SubMnd‘𝑈)))
Theorem	isdomn4 20661*	A ring is a domain iff it is nonzero and the left cancellation law for multiplication holds. (Contributed by SN, 15-Sep-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ (𝐵 ∖ { 0 })∀𝑏 ∈ 𝐵 ∀𝑐 ∈ 𝐵 ((𝑎 · 𝑏) = (𝑎 · 𝑐) → 𝑏 = 𝑐)))
Theorem	opprdomnb 20662	A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 20663. (Contributed by SN, 15-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn)
Theorem	opprdomn 20663	The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn)
Theorem	isdomn4r 20664*	A ring is a domain iff it is nonzero and the right cancellation law for multiplication holds. (Contributed by SN, 20-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ∀𝑐 ∈ (𝐵 ∖ { 0 })((𝑎 · 𝑐) = (𝑏 · 𝑐) → 𝑎 = 𝑏)))
Theorem	domnlcanb 20665	Left-cancellation law for domains, biconditional version of domnlcan 20666. (Contributed by Thierry Arnoux, 8-Jun-2025.) Shorten this theorem and domnlcan 20666 overall. (Revised by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) = (𝑋 · 𝑍) ↔ 𝑌 = 𝑍))
Theorem	domnlcan 20666	Left-cancellation law for domains. (Contributed by Thierry Arnoux, 22-Mar-2025.) (Proof shortened by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → (𝑋 · 𝑌) = (𝑋 · 𝑍))    ⇒   ⊢ (𝜑 → 𝑌 = 𝑍)
Theorem	domnrcanb 20667	Right-cancellation law for domains, biconditional version of domnrcan 20668. (Contributed by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑍) = (𝑌 · 𝑍) ↔ 𝑋 = 𝑌))
Theorem	domnrcan 20668	Right-cancellation law for domains. (Contributed by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    &   ⊢ (𝜑 → (𝑋 · 𝑍) = (𝑌 · 𝑍))    ⇒   ⊢ (𝜑 → 𝑋 = 𝑌)
Theorem	domneq0r 20669	Right multiplication by a nonzero element does not change zeroness in a domain. Compare rrgeq0 20645. (Contributed by SN, 21-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ (𝐵 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑅 ∈ Domn)    ⇒   ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 ))
Theorem	isidom 20670	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
Theorem	idomdomd 20671	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Domn)
Theorem	idomcringd 20672	An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) Formerly subproof of idomringd 20673. (Proof shortened by SN, 14-May-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ CRing)
Theorem	idomringd 20673	An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Ring)
10.4  Division rings and fields
10.4.1  Definition and basic properties
Syntax	cdr 20674	Extend class notation with class of all division rings.
class DivRing
Syntax	cfield 20675	Class of fields.
class Field
Definition	df-drng 20676	Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
⊢ DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g‘𝑟)})}
Definition	df-field 20677	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ Field = (DivRing ∩ CRing)
Theorem	isdrng 20678	The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))
Theorem	drngunit 20679	Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 )))
Theorem	drngui 20680	The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝑅 ∈ DivRing    ⇒   ⊢ (𝐵 ∖ { 0 }) = (Unit‘𝑅)
Theorem	drngring 20681	A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Ring)
Theorem	drngringd 20682	A division ring is a ring. (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Ring)
Theorem	drnggrpd 20683	A division ring is a group (deduction form). (Contributed by SN, 16-May-2024.)
⊢ (𝜑 → 𝑅 ∈ DivRing)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Grp)
Theorem	drnggrp 20684	A division ring is a group (closed form). (Contributed by NM, 8-Sep-2011.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Grp)
Theorem	isfld 20685	A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
Theorem	flddrngd 20686	A field is a division ring. (Contributed by SN, 17-Jan-2025.)
⊢ (𝜑 → 𝑅 ∈ Field)    ⇒   ⊢ (𝜑 → 𝑅 ∈ DivRing)
Theorem	fldcrngd 20687	A field is a commutative ring. (Contributed by SN, 23-Nov-2024.)
⊢ (𝜑 → 𝑅 ∈ Field)    ⇒   ⊢ (𝜑 → 𝑅 ∈ CRing)
Theorem	isdrng2 20688	A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))
Theorem	drngprop 20689	If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
⊢ (Base‘𝐾) = (Base‘𝐿)    &   ⊢ (+g‘𝐾) = (+g‘𝐿)    &   ⊢ (.r‘𝐾) = (.r‘𝐿)    ⇒   ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)
Theorem	drngmgp 20690	A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing → 𝐺 ∈ Grp)
Theorem	drngid 20691	A division ring's unity is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))    ⇒   ⊢ (𝑅 ∈ DivRing → 1 = (0g‘𝐺))
Theorem	drngunz 20692	A division ring's unity is different from its zero. (Contributed by NM, 8-Sep-2011.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → 1 ≠ 0 )
Theorem	drngnzr 20693	A division ring is a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
Theorem	drngdomn 20694	A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
⊢ (𝑅 ∈ DivRing → 𝑅 ∈ Domn)
Theorem	drngmcl 20695	The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) (Proof shortened by SN, 25-Jun-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 }))
Theorem	drngmclOLD 20696	Obsolete version of drngmcl 20695 as of 25-Jun-2025. The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 }))
Theorem	drngid2 20697	Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ DivRing → ((𝐼 ∈ 𝐵 ∧ 𝐼 ≠ 0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼))
Theorem	drnginvrcl 20698	Closure of the multiplicative inverse in a division ring. (reccl 11815 analog). (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ∈ 𝐵)
Theorem	drnginvrn0 20699	The multiplicative inverse in a division ring is nonzero. (recne0 11821 analog). (Contributed by NM, 19-Apr-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) → (𝐼‘𝑋) ≠ 0 )
Theorem	drnginvrcld 20700	Closure of the multiplicative inverse in a division ring. (reccld 11922 analog). (Contributed by SN, 14-Aug-2024.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ DivRing)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝐼‘𝑋) ∈ 𝐵)