P. 143 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14201-14300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isnzr2 14201	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	opprnzrbg 14202	The opposite of a nonzero ring is nonzero, bidirectional form of opprnzr 14203. (Contributed by SN, 20-Jun-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ NzRing ↔ 𝑂 ∈ NzRing))
Theorem	opprnzr 14203	The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
Theorem	ringelnzr 14204	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 14205	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	01eq0ring 14206	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 })
7.3.10  Local rings
Syntax	clring 14207	Extend class notation with class of all local rings.
class LRing
Definition	df-lring 14208*	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 14209*	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 14210	A local ring is a nonzero ring. (Contributed by SN, 23-Feb-2025.)
⊢ (𝑅 ∈ LRing → 𝑅 ∈ NzRing)
Theorem	lringring 14211	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 14212	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 14213	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))
7.3.11  Subrings
7.3.11.1  Subrings of non-unital rings
Syntax	csubrng 14214	Extend class notation with all subrings of a non-unital ring.
class SubRng
Definition	df-subrng 14215*	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 14216	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 14217	A subring is a subset. (Contributed by AV, 14-Feb-2025.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵)
Theorem	subrngid 14218	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 14219	A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng)
Theorem	subrngrcl 14220	Reverse closure for a subring predicate. (Contributed by AV, 14-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑅 ∈ Rng)
Theorem	subrngsubg 14221	A subring is a subgroup. (Contributed by AV, 14-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Theorem	subrngringnsg 14222	A subring is a normal subgroup. (Contributed by AV, 25-Feb-2025.)
⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ∈ (NrmSGrp‘𝑅))
Theorem	subrngbas 14223	Base set of a subring structure. (Contributed by AV, 14-Feb-2025.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 = (Base‘𝑆))
Theorem	subrng0 14224	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 14225	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 14226	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.) Generalization of subrgmcl 14250. (Revised by AV, 14-Feb-2025.)
⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRng‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
Theorem	issubrng2 14227*	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	opprsubrngg 14228	Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (SubRng‘𝑅) = (SubRng‘𝑂))
Theorem	subrngintm 14229*	The intersection of a nonempty collection of subrings is a subring. (Contributed by AV, 15-Feb-2025.)
⊢ ((𝑆 ⊆ (SubRng‘𝑅) ∧ ∃𝑗 𝑗 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRng‘𝑅))
Theorem	subrngin 14230	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	subsubrng 14231	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 14232	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	subrngpropd 14233*	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))
7.3.11.2  Subrings of unital rings
Syntax	csubrg 14234	Extend class notation with all subrings of a ring.
class SubRing
Syntax	crgspn 14235	Extend class notation with span of a set of elements over a ring.
class RingSpan
Definition	df-subrg 14236*	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 14237*	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 14238	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 14239	A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝐵 = (Base‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ⊆ 𝐵)
Theorem	subrgid 14240	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 14241	A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 𝑆 = (𝑅 ↾s 𝐴)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
Theorem	subrgcrng 14242	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 14243	Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
Theorem	subrgsubg 14244	A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
⊢ (𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
Theorem	subrg0 14245	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 14246	A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝐴 ∈ (SubRing‘𝑅) → 1 ∈ 𝐴)
Theorem	subrgbas 14247	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 14248	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 14249	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 14250	A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
Theorem	subrgsubm 14251	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 14252	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 14253	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 14254	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 14255	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 14256	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 14257	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 14258*	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	subrgnzr 14259	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	subrgintm 14260*	The intersection of an inhabited collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
⊢ ((𝑆 ⊆ (SubRing‘𝑅) ∧ ∃𝑤 𝑤 ∈ 𝑆) → ∩ 𝑆 ∈ (SubRing‘𝑅))
Theorem	subrgin 14261	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	subsubrg 14262	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 14263	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 14264	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 14265	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 14266	Restriction of the codomain of a (ring) homomorphism. resghm2b 13851 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 14267	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 14268	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 14269	The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
⊢ (𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
Theorem	subrgpropd 14270*	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 14271*	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))
7.3.12  Left regular elements and domains
Syntax	crlreg 14272	Set of left-regular elements in a ring.
class RLReg
Syntax	cdomn 14273	Class of (ring theoretic) domains.
class Domn
Syntax	cidom 14274	Class of integral domains.
class IDomn
Definition	df-rlreg 14275*	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 14276*	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 14277	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ IDomn = (CRing ∩ Domn)
Theorem	rrgmex 14278	A structure whose set of left-regular elements is inhabited is a set. (Contributed by Jim Kingdon, 12-Aug-2025.)
⊢ 𝐸 = (RLReg‘𝑅)    ⇒   ⊢ (𝐴 ∈ 𝐸 → 𝑅 ∈ V)
Theorem	rrgval 14279*	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 14280*	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 14281	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 14282	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	rrgss 14283	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 14284	Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
⊢ 𝐸 = (RLReg‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    ⇒   ⊢ (𝑅 ∈ Ring → 𝑈 ⊆ 𝐸)
Theorem	rrgnz 14285	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 14286*	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 14287	A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing)
Theorem	domnring 14288	A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring)
Theorem	domneq0 14289	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 14290	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	opprdomnbg 14291	A class is a domain if and only if its opposite is a domain, biconditional form of opprdomn 14292. (Contributed by SN, 15-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑅 ∈ Domn ↔ 𝑂 ∈ Domn))
Theorem	opprdomn 14292	The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
⊢ 𝑂 = (oppr‘𝑅)    ⇒   ⊢ (𝑅 ∈ Domn → 𝑂 ∈ Domn)
Theorem	isidom 14293	An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
⊢ (𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
Theorem	idomdomd 14294	An integral domain is a domain. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Domn)
Theorem	idomcringd 14295	An integral domain is a commutative ring with unity. (Contributed by Thierry Arnoux, 4-May-2025.) (Proof shortened by SN, 14-May-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ CRing)
Theorem	idomringd 14296	An integral domain is a ring. (Contributed by Thierry Arnoux, 22-Mar-2025.)
⊢ (𝜑 → 𝑅 ∈ IDomn)    ⇒   ⊢ (𝜑 → 𝑅 ∈ Ring)
7.4  Division rings and fields
7.4.1  Ring apartness
Syntax	capr 14297	Extend class notation with ring apartness.
class #r
Definition	df-apr 14298*	The relation between elements whose difference is invertible, which for a local ring is an apartness relation by aprap 14303. (Contributed by Jim Kingdon, 13-Feb-2025.)
⊢ #r = (𝑤 ∈ V ↦ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (Base‘𝑤) ∧ 𝑦 ∈ (Base‘𝑤)) ∧ (𝑥(-g‘𝑤)𝑦) ∈ (Unit‘𝑤))})
Theorem	aprval 14299	Expand Definition df-apr 14298. (Contributed by Jim Kingdon, 17-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → # = (#r‘𝑅))    &   ⊢ (𝜑 → − = (-g‘𝑅))    &   ⊢ (𝜑 → 𝑈 = (Unit‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 # 𝑌 ↔ (𝑋 − 𝑌) ∈ 𝑈))
Theorem	aprirr 14300	The apartness relation given by df-apr 14298 for a nonzero ring is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2025.)
⊢ (𝜑 → 𝐵 = (Base‘𝑅))    &   ⊢ (𝜑 → # = (#r‘𝑅))    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → (1r‘𝑅) ≠ (0g‘𝑅))    ⇒   ⊢ (𝜑 → ¬ 𝑋 # 𝑋)