P. 205 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rhmopp 20401	A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
⊢ (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ ((oppr‘𝑅) RingHom (oppr‘𝑆)))
Theorem	elrhmunit 20402	Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘𝐴) ∈ (Unit‘𝑆))
Theorem	rhmunitinv 20403	Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
⊢ ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴 ∈ (Unit‘𝑅)) → (𝐹‘((invr‘𝑅)‘𝐴)) = ((invr‘𝑆)‘(𝐹‘𝐴)))
10.3.11  Nonzero rings and zero rings
Syntax	cnzr 20404	The class of nonzero rings.
class NzRing
Definition	df-nzr 20405	A nonzero or nontrivial ring is a ring with at least two values, or equivalently where 1 and 0 are different. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ NzRing = {𝑟 ∈ Ring ∣ (1r‘𝑟) ≠ (0g‘𝑟)}
Theorem	isnzr 20406	Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 ≠ 0 ))
Theorem	nzrnz 20407	One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 1 ≠ 0 )
Theorem	nzrring 20408	A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof shortened by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Theorem	nzrringOLD 20409	Obsolete version of nzrring 20408 as of 23-Feb-2025. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring)
Theorem	isnzr2 20410	Equivalent characterization of nonzero rings: they have at least two elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 2o ≼ 𝐵))
Theorem	isnzr2hash 20411	Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20410. (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)))
Theorem	opprnzr 20412	The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
Theorem	ringelnzr 20413	A ring is nonzero if it has a nonzero element. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by Mario Carneiro, 13-Jun-2015.)
⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing)
Theorem	nzrunit 20414	A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → 𝐴 ≠ 0 )
Theorem	0ringnnzr 20415	A ring is a zero ring iff it is not a nonzero ring. (Contributed by AV, 14-Apr-2019.)
⊢ (𝑅 ∈ Ring → ((♯‘(Base‘𝑅)) = 1 ↔ ¬ 𝑅 ∈ NzRing))
Theorem	0ring 20416	If a ring has only one element, it is the zero ring. According to Wikipedia ("Zero ring", 14-Apr-2019, https://en.wikipedia.org/wiki/Zero_ring): "The zero ring, denoted {0} or simply 0, consists of the one-element set {0} with the operations + and * defined so that 0 + 0 = 0 and 0 * 0 = 0.". (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 })
Theorem	0ringdif 20417	A zero ring is a ring which is not a nonzero ring. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ (Ring ∖ NzRing) ↔ (𝑅 ∈ Ring ∧ 𝐵 = { 0 }))
Theorem	0ringbas 20418	The base set of a zero ring, a ring which is not a nonzero ring, is the singleton of the zero element. (Contributed by AV, 17-Apr-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝑅 ∈ (Ring ∖ NzRing) → 𝐵 = { 0 })
Theorem	0ring01eq 20419	In a ring with only one element, i.e. a zero ring, the zero and the identity element are the same. (Contributed by AV, 14-Apr-2019.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 )
Theorem	01eq0ring 20420	If the zero and the identity element of a ring are the same, the ring is the zero ring. (Contributed by AV, 16-Apr-2019.) (Proof shortened by SN, 23-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
Theorem	01eq0ringOLD 20421	Obsolete version of 01eq0ring 20420 as of 23-Feb-2025. (Contributed by AV, 16-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
Theorem	0ring01eqbi 20422	In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (Revised by AV, 23-Jan-2020.)
⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → (𝐵 ≈ 1o ↔ 1 = 0 ))
Theorem	0ring1eq0 20423	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 20424*	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 20425*	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 20426*	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))
10.3.12  Local rings
Syntax	clring 20427	Extend class notation with class of all local rings.
class LRing
Definition	df-lring 20428*	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 20429*	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 20430	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
Theorem	lringring 20431	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 20432	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 20433	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 20434	Extend class notation with all subrings of a non-unital ring.
class SubRng
Definition	df-subrng 20435*	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 20436	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 20437	A subring is a subset. (Contributed by AV, 14-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
Theorem	subrngid 20438	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 20439	A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
Theorem	subrngrcl 20440	Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
Theorem	subrngsubg 20441	A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Theorem	subrngringnsg 20442	A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
Theorem	subrngbas 20443	Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
Theorem	subrng0 20444	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 20445	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 20446	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 20475. (Revised by AV, 14-Feb-2025.)
⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
Theorem	issubrng2 20447*	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 20448	Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (SubRng‘𝑅) = (SubRng‘𝑂)
Theorem	subrngint 20449	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 20450	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 20451	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 20452	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 20453	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 20454*	Lemma for rhmimasubrng 20455: Modified part of mhmima 18743. (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 20455	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 20456	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 20457*	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 20458	Extend class notation with all subrings of a ring.
class SubRing
Syntax	crgspn 20459	Extend class notation with span of a set of elements over a ring.
class RingSpan
Definition	df-subrg 20460*	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)})
Definition	df-rgspn 20461*	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	issubrg 20462	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 20463	A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵)
Theorem	subrgid 20464	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 20465	A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
Theorem	subrgcrng 20466	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 20467	Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Theorem	subrgsubg 20468	A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Theorem	subrgsubrng 20469	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 20470	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 20471	A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴)
Theorem	subrgbas 20472	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 20473	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 20474	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 20475	A subgroup 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 20476	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 20477	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 20478	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 20479	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 20480	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 20481	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 20482	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 20483*	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 20484	Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (SubRing‘𝑅) = (SubRing‘𝑂)
Theorem	subrgnzr 20485	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 20486	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 20487	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 20488	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 20489	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 20490	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 20491	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 20492	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 20493	Restriction of the codomain of a (ring) homomorphism. resghm2b 19149 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 20494	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 20495	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 20496	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 20497	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 20498*	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 20499*	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 20500*	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))