P. 208 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	0ring1eq0 20701	In a zero ring, a ring which is not a nonzero ring, the ring unity equals the zero element. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 1 = 0 )
Theorem	c0rhm 20702*	The constant mapping to zero is a ring homomorphism from any ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Ring ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RingHom 𝑇))
Theorem	c0rnghm 20703*	The constant mapping to zero is a non-unital ring homomorphism from any non-unital ring to the zero ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑆 RngHom 𝑇))
Theorem	zrrnghm 20704*	The constant mapping to zero is a non-unital ring homomorphism from the zero ring to any non-unital ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ 𝐻 = (𝑥 ∈ 𝐵 ↦ 0 )    ⇒   ⊢ ((𝑆 ∈ Rng ∧ 𝑇 ∈ (Ring ∖ NzRing)) → 𝐻 ∈ (𝑇 RngHom 𝑆))
Theorem	nrhmzr 20705	There is no ring homomorphism from the zero ring into a nonzero ring. (Contributed by AV, 18-Apr-2020.)
⊢ ((𝑍 ∈ (Ring ∖ NzRing) ∧ 𝑅 ∈ NzRing) → (𝑍 RingHom 𝑅) = ∅)
10.3.12  Local rings
Syntax	clring 20706	Extend class notation with class of all local rings.
class LRing
Definition	df-lring 20707*	A local ring is a nonzero ring where for any two elements summing to one, at least one is invertible. Any field is a local ring; the ring of integers is an example of a ring which is not a local ring. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ LRing = {𝑟 ∈ NzRing ∣ ∀𝑥 ∈ (Base‘𝑟)∀𝑦 ∈ (Base‘𝑟)((𝑥(+g‘𝑟)𝑦) = (1r‘𝑟) → (𝑥 ∈ (Unit‘𝑟) ∨ 𝑦 ∈ (Unit‘𝑟)))}
Theorem	islring 20708*	The predicate "is a local ring". (Contributed by SN, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ + = (+g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ LRing ↔ (𝑅 ∈ NzRing ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 1 → (𝑥 ∈ 𝑈 ∨ 𝑦 ∈ 𝑈))))
Theorem	lringnzr 20709	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
Theorem	lringring 20710	A local ring is a ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ Ring)
Theorem	lringnz 20711	A local ring is a nonzero ring. (Contributed by Jim Kingdon, 20-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ LRing → 1 ≠ 0 )
Theorem	lringuplu 20712	If the sum of two elements of a local ring is invertible, then at least one of the summands must be invertible. (Contributed by Jim Kingdon, 18-Feb-2025.) (Revised by SN, 23-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → + = (+g‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ LRing)    &   ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝑈)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝑈 ∨ 𝑌 ∈ 𝑈))
10.3.13  Subrings
10.3.13.1  Subrings of non-unital rings
Syntax	csubrng 20713	Extend class notation with all subrings of a non-unital ring.
class SubRng
Definition	df-subrng 20714*	Define a subring of a non-unital ring as a set of elements that is a non-unital ring in its own right. In this section, a subring of a non-unital ring is simply called "subring", unless it causes any ambiguity with SubRing. (Contributed by AV, 14-Feb-2025.)
⊢ SubRng = (𝑤 ∈ Rng ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ (𝑤 ↾s 𝑠) ∈ Rng})
Theorem	issubrng 20715	The subring of non-unital ring predicate. (Contributed by AV, 14-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵))
Theorem	subrngss 20716	A subring is a subset. (Contributed by AV, 14-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
Theorem	subrngid 20717	Every non-unital ring is a subring of itself. (Contributed by AV, 14-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → 𝐵 ∈ (SubRng‘𝑅))
Theorem	subrngrng 20718	A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
Theorem	subrngrcl 20719	Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
Theorem	subrngsubg 20720	A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Theorem	subrngringnsg 20721	A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
Theorem	subrngbas 20722	Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
Theorem	subrng0 20723	A subring always has the same additive identity. (Contributed by AV, 14-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 0 = (0g‘𝑆))
Theorem	subrngacl 20724	A subring is closed under addition. (Contributed by AV, 14-Feb-2025.)
⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
Theorem	subrngmcl 20725	A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 20752. (Revised by AV, 14-Feb-2025.)
⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
Theorem	issubrng2 20726*	Characterize the subrings of a ring by closure properties. (Contributed by AV, 15-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → (𝐴 ∈ (SubRng‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Theorem	opprsubrng 20727	Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (SubRng‘𝑅) = (SubRng‘𝑂)
Theorem	subrngint 20728	The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)
⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRng‘𝑅))
Theorem	subrngin 20729	The intersection of two subrings is a subring. (Contributed by AV, 15-Feb-2025.)
⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝐵 ∈ (SubRng‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRng‘𝑅))
Theorem	subrngmre 20730	The subrings of a non-unital ring are a Moore system. (Contributed by AV, 15-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Rng → (SubRng‘𝑅) ∈ (Moore‘𝐵))
Theorem	subsubrng 20731	A subring of a subring is a subring. (Contributed by AV, 15-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → (𝐵 ∈ (SubRng‘𝑆) ↔ (𝐵 ∈ (SubRng‘𝑅) ∧ 𝐵 ⊆ 𝐴)))
Theorem	subsubrng2 20732	The set of subrings of a subring are the smaller subrings. (Contributed by AV, 15-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → (SubRng‘𝑆) = ((SubRng‘𝑅) ∩ 𝒫 𝐴))
Theorem	rhmimasubrnglem 20733*	Lemma for rhmimasubrng 20734: Modified part of mhmima 18940. (Contributed by Mario Carneiro, 10-Mar-2015.) (Revised by AV, 16-Feb-2025.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑅)) → ∀𝑥 ∈ (𝐹 “ 𝑋)∀𝑦 ∈ (𝐹 “ 𝑋)(𝑥(+g‘𝑁)𝑦) ∈ (𝐹 “ 𝑋))
Theorem	rhmimasubrng 20734	The homomorphic image of a subring is a subring. (Contributed by AV, 16-Feb-2025.)
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRng‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRng‘𝑁))
Theorem	cntzsubrng 20735	Centralizers in a non-unital ring are subrings. (Contributed by AV, 17-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Rng ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRng‘𝑅))
Theorem	subrngpropd 20736*	If two structures have the same ring components (properties), they have the same set of subrings. (Contributed by AV, 17-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (SubRng‘𝐾) = (SubRng‘𝐿))
10.3.13.2  Subrings of unital rings
Syntax	csubrg 20737	Extend class notation with all subrings of a ring.
class SubRing
Definition	df-subrg 20738*	Define a subring of a ring as a set of elements that is a ring in its own right and contains the multiplicative identity. The additional constraint is necessary because the multiplicative identity of a ring, unlike the additive identity of a ring/group or the multiplicative identity of a field, cannot be identified by a local property. Thus, it is possible for a subset of a ring to be a ring while not containing the true identity if it contains a false identity. For instance, the subset (ℤ × {0}) of (ℤ × ℤ) (where multiplication is componentwise) contains the false identity ⟨1, 0⟩ which preserves every element of the subset and thus appears to be the identity of the subset, but is not the identity of the larger ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ SubRing = (𝑤 ∈ Ring ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ((𝑤 ↾s 𝑠) ∈ Ring ∧ (1r‘𝑤) ∈ 𝑠)})
Theorem	issubrg 20739	The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅 ↾s 𝐴) ∈ Ring) ∧ (𝐴 ⊆ 𝐵 ∧ 1 ∈ 𝐴)))
Theorem	subrgss 20740	A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵)
Theorem	subrgid 20741	Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
Theorem	subrgring 20742	A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
Theorem	subrgcrng 20743	A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ CRing)
Theorem	subrgrcl 20744	Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Theorem	subrgsubg 20745	A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Theorem	subrgsubrng 20746	A subring of a unital ring is a subring of a non-unital ring. (Contributed by AV, 30-Mar-2025.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubRng‘𝑅))
Theorem	subrg0 20747	A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g‘𝑆))
Theorem	subrg1cl 20748	A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴)
Theorem	subrgbas 20749	Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
Theorem	subrg1 20750	A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r‘𝑆))
Theorem	subrgacl 20751	A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ + = (+g‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
Theorem	subrgmcl 20752	A subring is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 30-Mar-2025.)
⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
Theorem	subrgsubm 20753	A subring is a submonoid of the multiplicative monoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubMnd‘𝑀))
Theorem	subrgdvds 20754	If an element divides another in a subring, then it also divides the other in the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ ∥ = (∥r‘𝑅)    &   ⊢ 𝐸 = (∥r‘𝑆)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐸 ⊆ ∥ )
Theorem	subrguss 20755	A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑉 = (Unit‘𝑆)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ⊆ 𝑈)
Theorem	subrginv 20756	A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑆)    &   ⊢ 𝐽 = (invr‘𝑆)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝑈) → (𝐼‘𝑋) = (𝐽‘𝑋))
Theorem	subrgdv 20757	A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ / = (/r‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑆)    &   ⊢ 𝐸 = (/r‘𝑆)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌))
Theorem	subrgunit 20758	An element of a ring is a unit of a subring iff it is a unit of the parent ring and both it and its inverse are in the subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑉 = (Unit‘𝑆)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝑋 ∈ 𝑉 ↔ (𝑋 ∈ 𝑈 ∧ 𝑋 ∈ 𝐴 ∧ (𝐼‘𝑋) ∈ 𝐴)))
Theorem	subrgugrp 20759	The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝑉 = (Unit‘𝑆)    &   ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))
Theorem	issubrg2 20760*	Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
Theorem	opprsubrg 20761	Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (SubRing‘𝑅) = (SubRing‘𝑂)
Theorem	subrgnzr 20762	A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing)
Theorem	subrgint 20763	The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ (SubRing‘𝑅))
Theorem	subrgin 20764	The intersection of two subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝐵 ∈ (SubRing‘𝑅)) → (𝐴 ∩ 𝐵) ∈ (SubRing‘𝑅))
Theorem	subrgmre 20765	The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (Moore‘𝐵))
Theorem	subsubrg 20766	A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵 ⊆ 𝐴)))
Theorem	subsubrg2 20767	The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → (SubRing‘𝑆) = ((SubRing‘𝑅) ∩ 𝒫 𝐴))
Theorem	issubrg3 20768	A subring is an additive subgroup which is also a multiplicative submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
⊢ 𝑀 = (mulGrp‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝑆 ∈ (SubRing‘𝑅) ↔ (𝑆 ∈ (SubGrp‘𝑅) ∧ 𝑆 ∈ (SubMnd‘𝑀))))
Theorem	resrhm 20769	Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
⊢ 𝑈 = (𝑆 ↾s 𝑋)    ⇒   ⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹 ↾ 𝑋) ∈ (𝑈 RingHom 𝑇))
Theorem	resrhm2b 20770	Restriction of the codomain of a (ring) homomorphism. resghm2b 19367 analog. (Contributed by SN, 7-Feb-2025.)
⊢ 𝑈 = (𝑇 ↾s 𝑋)    ⇒   ⊢ ((𝑋 ∈ (SubRing‘𝑇) ∧ ran 𝐹 ⊆ 𝑋) → (𝐹 ∈ (𝑆 RingHom 𝑇) ↔ 𝐹 ∈ (𝑆 RingHom 𝑈)))
Theorem	rhmeql 20771	The equalizer of two ring homomorphisms is a subring. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → dom (𝐹 ∩ 𝐺) ∈ (SubRing‘𝑆))
Theorem	rhmima 20772	The homomorphic image of a subring is a subring. (Contributed by Stefan O'Rear, 10-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ ((𝐹 ∈ (𝑀 RingHom 𝑁) ∧ 𝑋 ∈ (SubRing‘𝑀)) → (𝐹 “ 𝑋) ∈ (SubRing‘𝑁))
Theorem	rnrhmsubrg 20773	The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
Theorem	cntzsubr 20774	Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑀 = (mulGrp‘𝑅)    &   ⊢ 𝑍 = (Cntz‘𝑀)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑆 ⊆ 𝐵) → (𝑍‘𝑆) ∈ (SubRing‘𝑅))
Theorem	pwsdiagrhm 20775*	Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝑌 = (𝑅 ↑s 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝐼 × {𝑥}))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ (𝑅 RingHom 𝑌))
Theorem	subrgpropd 20776*	If two structures have the same group components (properties), they have the same set of subrings. (Contributed by Mario Carneiro, 9-Feb-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦))    ⇒   ⊢ (𝜑 → (SubRing‘𝐾) = (SubRing‘𝐿))
Theorem	rhmpropd 20777*	Ring homomorphism depends only on the ring attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
⊢ (𝜑 → 𝐵 = (Base‘𝐽))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝐾))    &   ⊢ (𝜑 → 𝐵 = (Base‘𝐿))    &   ⊢ (𝜑 → 𝐶 = (Base‘𝑀))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐽)𝑦) = (𝑥(.r‘𝐿)𝑦))    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝑀)𝑦))    ⇒   ⊢ (𝜑 → (𝐽 RingHom 𝐾) = (𝐿 RingHom 𝑀))
10.3.13.3  Subrings generated by a subset
Syntax	crgspn 20778	Extend class notation with span of a set of elements over a ring.
class RingSpan
Definition	df-rgspn 20779*	The ring-span of a set of elements in a ring is the smallest subring which contains all of them. (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ RingSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ ∩ {𝑡 ∈ (SubRing‘𝑤) ∣ 𝑠 ⊆ 𝑡}))
Theorem	rgspnval 20780*	Value of the ring-span of a set of elements in a ring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴))    ⇒   ⊢ (𝜑 → 𝑈 = ∩ {𝑡 ∈ (SubRing‘𝑅) ∣ 𝐴 ⊆ 𝑡})
Theorem	rgspncl 20781	The ring-span of a set is a subring. (Contributed by Stefan O'Rear, 7-Dec-2014.)
⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴))    ⇒   ⊢ (𝜑 → 𝑈 ∈ (SubRing‘𝑅))
Theorem	rgspnssid 20782	The ring-span of a set contains the set. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴))    ⇒   ⊢ (𝜑 → 𝐴 ⊆ 𝑈)
Theorem	rgspnmin 20783	The ring-span is contained in all subrings which contain all the generators. (Contributed by Stefan O'Rear, 30-Nov-2014.)
⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝐵)    &   ⊢ (𝜑 → 𝑁 = (RingSpan‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (𝑁‘𝐴))    &   ⊢ (𝜑 → 𝑆 ∈ (SubRing‘𝑅))    &   ⊢ (𝜑 → 𝐴 ⊆ 𝑆)    ⇒   ⊢ (𝜑 → 𝑈 ⊆ 𝑆)
10.3.14  Categories of rings
10.3.14.1  The category of non-unital rings The "category of non-unital rings" RngCat is the category of all non-unital rings Rng in a universe and non-unital ring homomorphisms RngHom between these rings. This category is defined as "category restriction" of the category of extensible structures ExtStrCat, which restricts the objects to non-unital rings and the morphisms to the non-unital ring homomorphisms, while the composition of morphisms is preserved, see df-rngc 20785. Alternately, the category of non-unital rings could have been defined as extensible structure consisting of three components/slots for the objects, morphisms and composition, see dfrngc2 20796. Since we consider only "small categories" (i.e. categories whose objects and morphisms are actually sets and not proper classes), the objects of the category (i.e. the base set of the category regarded as extensible structure) are a subset of the non-unital rings (relativized to a subset or "universe" 𝑢) (𝑢 ∩ Rng), see rngcbas 20789, and the morphisms/arrows are the non-unital ring homomorphisms restricted to this subset of the non-unital rings ( RngHom ↾ (𝐵 × 𝐵)), see rngchomfval 20790, whereas the composition is the ordinary composition of functions, see rngccofval 20794 and rngcco 20795. By showing that the non-unital ring homomorphisms between non-unital rings are a subcategory subset (⊆cat) of the mappings between base sets of extensible structures, see rnghmsscmap 20798, it can be shown that the non-unital ring homomorphisms between non-unital rings are a subcategory (Subcat) of the category of extensible structures, see rnghmsubcsetc 20801. It follows that the category of non-unital rings RngCat is actually a category, see rngccat 20802 with the identity function as identity arrow, see rngcid 20803.
Syntax	crngc 20784	Extend class notation to include the category Rng.
class RngCat
Definition	df-rngc 20785	Definition of the category Rng, relativized to a subset 𝑢. This is the category of all non-unital rings in 𝑢 and homomorphisms between these rings. Generally, we will take 𝑢 to be a weak universe or Grothendieck universe, because these sets have closure properties as good as the real thing. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
Theorem	rngcval 20786	Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))    &   ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Theorem	rnghmresfn 20787	The class of non-unital ring homomorphisms restricted to subsets of non-unital rings is a function. (Contributed by AV, 4-Mar-2020.)
⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))    &   ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ (𝜑 → 𝐻 Fn (𝐵 × 𝐵))
Theorem	rnghmresel 20788	An element of the non-unital ring homomorphisms restricted to a subset of non-unital rings is a non-unital ring homomorphisms. (Contributed by AV, 9-Mar-2020.)
⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 ∈ (𝑋 RngHom 𝑌))
Theorem	rngcbas 20789	Set of objects of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
Theorem	rngchomfval 20790	Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    ⇒   ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
Theorem	rngchom 20791	Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ 𝐻 = (Hom ‘𝐶)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋 RngHom 𝑌))
Theorem	elrngchom 20792	A morphism of non-unital rings is a function. (Contributed by AV, 27-Feb-2020.)
⊢ 𝐶 = (RngCat‘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	rngchomfeqhom 20793	The functionalized Hom-set operation equals the Hom-set operation in the category of non-unital rings (in a universe). (Contributed by AV, 9-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ 𝐵 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (Homf ‘𝐶) = (Hom ‘𝐶))
Theorem	rngccofval 20794	Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ · = (comp‘𝐶)    ⇒   ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))
Theorem	rngcco 20795	Composition in the category of non-unital rings. (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
⊢ 𝐶 = (RngCat‘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	dfrngc2 20796	Alternate definition of the category of non-unital rings (in a universe). (Contributed by AV, 16-Mar-2020.)
⊢ 𝐶 = (RngCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))    &   ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))    &   ⊢ (𝜑 → · = (comp‘(ExtStrCat‘𝑈)))    ⇒   ⊢ (𝜑 → 𝐶 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Theorem	rnghmsscmap2 20797*	The non-unital ring homomorphisms between non-unital rings (in a universe) are a subcategory subset of the mappings between base sets of non-unital rings (in the same universe). (Contributed by AV, 6-Mar-2020.)
⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 = (Rng ∩ 𝑈))    ⇒   ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑅 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Theorem	rnghmsscmap 20798*	The non-unital ring homomorphisms between non-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 = (Rng ∩ 𝑈))    ⇒   ⊢ (𝜑 → ( RngHom ↾ (𝑅 × 𝑅)) ⊆cat (𝑥 ∈ 𝑈, 𝑦 ∈ 𝑈 ↦ ((Base‘𝑦) ↑m (Base‘𝑥))))
Theorem	rnghmsubcsetclem1 20799	Lemma 1 for rnghmsubcsetc 20801. (Contributed by AV, 9-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((Id‘𝐶)‘𝑥) ∈ (𝑥𝐻𝑥))
Theorem	rnghmsubcsetclem2 20800*	Lemma 2 for rnghmsubcsetc 20801. (Contributed by AV, 9-Mar-2020.)
⊢ 𝐶 = (ExtStrCat‘𝑈)    &   ⊢ (𝜑 → 𝑈 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐵 = (Rng ∩ 𝑈))    &   ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))    ⇒   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑓 ∈ (𝑥𝐻𝑦)∀𝑔 ∈ (𝑦𝐻𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) ∈ (𝑥𝐻𝑧))