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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlidlacs 20501 The ideal system is an algebraic closure system on the base set. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐵 = (Base‘𝑊)    &   𝐼 = (LIdeal‘𝑊)       (𝑊 ∈ Ring → 𝐼 ∈ (ACS‘𝐵))
 
Theoremrspcl 20502 The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → (𝐾𝐺) ∈ 𝑈)
 
Theoremrspssid 20503 The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → 𝐺 ⊆ (𝐾𝐺))
 
Theoremrsp1 20504 The span of the identity element is the unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 1 }) = 𝐵)
 
Theoremrsp0 20505 The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → (𝐾‘{ 0 }) = { 0 })
 
Theoremrspssp 20506 The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐾 = (RSpan‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐺𝐼) → (𝐾𝐺) ⊆ 𝐼)
 
Theoremmrcrsp 20507 Moore closure generalizes ideal span. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐹 = (mrCls‘𝑈)       (𝑅 ∈ Ring → 𝐾 = 𝐹)
 
Theoremlidlnz 20508* A nonzero ideal contains a nonzero element. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑈 = (LIdeal‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐼 ≠ { 0 }) → ∃𝑥𝐼 𝑥0 )
 
Theoremdrngnidl 20509 A division ring has only the two trivial ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ DivRing → 𝑈 = {{ 0 }, 𝐵})
 
Theoremlidlrsppropd 20510* The left ideals and ring span of a ring depend only on the ring components. Here 𝑊 is expected to be either 𝐵 (when closure is available) or V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → ((LIdeal‘𝐾) = (LIdeal‘𝐿) ∧ (RSpan‘𝐾) = (RSpan‘𝐿)))
 
10.7.2  Two-sided ideals and quotient rings
 
Syntaxc2idl 20511 Ring two-sided ideal function.
class 2Ideal
 
Definitiondf-2idl 20512 Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
2Ideal = (𝑟 ∈ V ↦ ((LIdeal‘𝑟) ∩ (LIdeal‘(oppr𝑟))))
 
Theorem2idlval 20513 Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)    &   𝐽 = (LIdeal‘𝑂)    &   𝑇 = (2Ideal‘𝑅)       𝑇 = (𝐼𝐽)
 
Theorem2idlcpbl 20514 The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑋 = (Base‘𝑅)    &   𝐸 = (𝑅 ~QG 𝑆)    &   𝐼 = (2Ideal‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → ((𝐴𝐸𝐶𝐵𝐸𝐷) → (𝐴 · 𝐵)𝐸(𝐶 · 𝐷)))
 
Theoremqus1 20515 The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → (𝑈 ∈ Ring ∧ [ 1 ](𝑅 ~QG 𝑆) = (1r𝑈)))
 
Theoremqusring 20516 If 𝑆 is a two-sided ideal in 𝑅, then 𝑈 = 𝑅 / 𝑆 is a ring, called the quotient ring of 𝑅 by 𝑆. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → 𝑈 ∈ Ring)
 
Theoremqusrhm 20517* If 𝑆 is a two-sided ideal in 𝑅, then the "natural map" from elements to their cosets is a ring homomorphism from 𝑅 to 𝑅 / 𝑆. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (2Ideal‘𝑅)    &   𝑋 = (Base‘𝑅)    &   𝐹 = (𝑥𝑋 ↦ [𝑥](𝑅 ~QG 𝑆))       ((𝑅 ∈ Ring ∧ 𝑆𝐼) → 𝐹 ∈ (𝑅 RingHom 𝑈))
 
Theoremcrngridl 20518 In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)    &   𝑂 = (oppr𝑅)       (𝑅 ∈ CRing → 𝐼 = (LIdeal‘𝑂))
 
Theoremcrng2idl 20519 In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
𝐼 = (LIdeal‘𝑅)       (𝑅 ∈ CRing → 𝐼 = (2Ideal‘𝑅))
 
Theoremquscrng 20520 The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑈 = (𝑅 /s (𝑅 ~QG 𝑆))    &   𝐼 = (LIdeal‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑆𝐼) → 𝑈 ∈ CRing)
 
10.7.3  Principal ideal rings. Divisibility in the integers
 
Syntaxclpidl 20521 Ring left-principal-ideal function.
class LPIdeal
 
Syntaxclpir 20522 Class of left principal ideal rings.
class LPIR
 
Definitiondf-lpidl 20523* Define the class of left principal ideals of a ring, which are ideals with a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIdeal = (𝑤 ∈ Ring ↦ 𝑔 ∈ (Base‘𝑤){((RSpan‘𝑤)‘{𝑔})})
 
Definitiondf-lpir 20524 Define the class of left principal ideal rings, rings where every left ideal has a single generator. (Contributed by Stefan O'Rear, 3-Jan-2015.)
LPIR = {𝑤 ∈ Ring ∣ (LIdeal‘𝑤) = (LPIdeal‘𝑤)}
 
Theoremlpival 20525* Value of the set of principal ideals. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝑃 = 𝑔𝐵 {(𝐾‘{𝑔})})
 
Theoremislpidl 20526* Property of being a principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (𝐼𝑃 ↔ ∃𝑔𝐵 𝐼 = (𝐾‘{𝑔})))
 
Theoremlpi0 20527 The zero ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Ring → { 0 } ∈ 𝑃)
 
Theoremlpi1 20528 The unit ideal is always principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵𝑃)
 
Theoremislpir 20529 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈 = 𝑃))
 
Theoremlpiss 20530 Principal ideals are a subclass of ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ Ring → 𝑃𝑈)
 
Theoremislpir2 20531 Principal ideal rings are where all ideals are principal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝑃 = (LPIdeal‘𝑅)    &   𝑈 = (LIdeal‘𝑅)       (𝑅 ∈ LPIR ↔ (𝑅 ∈ Ring ∧ 𝑈𝑃))
 
Theoremlpirring 20532 Principal ideal rings are rings. (Contributed by Stefan O'Rear, 24-Jan-2015.)
(𝑅 ∈ LPIR → 𝑅 ∈ Ring)
 
Theoremdrnglpir 20533 Division rings are principal ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ LPIR)
 
Theoremrspsn 20534* Membership in principal ideals is closely related to divisibility. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐺𝐵) → (𝐾‘{𝐺}) = {𝑥𝐺 𝑥})
 
Theoremlidldvgen 20535* An element generates an ideal iff it is contained in the ideal and all elements are right-divided by it. (Contributed by Stefan O'Rear, 3-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (LIdeal‘𝑅)    &   𝐾 = (RSpan‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈𝐺𝐵) → (𝐼 = (𝐾‘{𝐺}) ↔ (𝐺𝐼 ∧ ∀𝑥𝐼 𝐺 𝑥)))
 
Theoremlpigen 20536* An ideal is principal iff it contains an element which right-divides all elements. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Wolf Lammen, 6-Sep-2020.)
𝑈 = (LIdeal‘𝑅)    &   𝑃 = (LPIdeal‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑈) → (𝐼𝑃 ↔ ∃𝑥𝐼𝑦𝐼 𝑥 𝑦))
 
10.7.4  Nonzero rings and zero rings
 
Syntaxcnzr 20537 The class of nonzero rings.
class NzRing
 
Definitiondf-nzr 20538 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 ∣ (1r𝑟) ≠ (0g𝑟)}
 
Theoremisnzr 20539 Property of a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 10 ))
 
Theoremnzrnz 20540 One and zero are different in a nonzero ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
1 = (1r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ NzRing → 10 )
 
Theoremnzrring 20541 A nonzero ring is a ring. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ NzRing → 𝑅 ∈ Ring)
 
Theoremdrngnzr 20542 All division rings are nonzero. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
 
Theoremisnzr2 20543 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 ∧ 2o𝐵))
 
Theoremisnzr2hash 20544 Equivalent characterization of nonzero rings: they have at least two elements. Analogous to isnzr2 20543. (Contributed by AV, 14-Apr-2019.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ NzRing ↔ (𝑅 ∈ Ring ∧ 1 < (♯‘𝐵)))
 
Theoremopprnzr 20545 The opposite of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 17-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ NzRing → 𝑂 ∈ NzRing)
 
Theoremringelnzr 20546 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 = (0g𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵 ∖ { 0 })) → 𝑅 ∈ NzRing)
 
Theoremnzrunit 20547 A unit is nonzero in any nonzero ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ NzRing ∧ 𝐴𝑈) → 𝐴0 )
 
Theoremsubrgnzr 20548 A subring of a nonzero ring is nonzero. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑆 = (𝑅s 𝐴)       ((𝑅 ∈ NzRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ NzRing)
 
Theorem0ringnnzr 20549 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))
 
Theorem0ring 20550 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 = (0g𝑅)       ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 𝐵 = { 0 })
 
Theorem0ring01eq 20551 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 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ (♯‘𝐵) = 1) → 0 = 1 )
 
Theorem01eq0ring 20552 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.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 0 = 1 ) → 𝐵 = { 0 })
 
Theorem0ring01eqbi 20553 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 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐵 ≈ 1o1 = 0 ))
 
Theoremrng1nnzr 20554 The (smallest) structure representing a zero ring is not a nonzero ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ NzRing)
 
Theoremring1zr 20555 The only (unital) ring with a base set consisting of one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 13-Feb-2010.) (Revised by AV, 25-Jan-2020.) (Proof shortened by AV, 7-Feb-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 = {𝑍} ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremrngen1zr 20556 The only (unital) ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 14-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)       (((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) ∧ 𝑍𝐵) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremringen1zr 20557 The only unital ring with one element is the zero ring (at least if its operations are internal binary operations). Note: The assumption 𝑅 ∈ Ring could be weakened if a definition of a non-unital ring ("Rng") was available (it would be sufficient that the multiplication is closed). (Contributed by FL, 15-Feb-2010.) (Revised by AV, 25-Jan-2020.)
𝐵 = (Base‘𝑅)    &    + = (+g𝑅)    &    = (.r𝑅)    &   𝑍 = (0g𝑅)       ((𝑅 ∈ Ring ∧ + Fn (𝐵 × 𝐵) ∧ Fn (𝐵 × 𝐵)) → (𝐵 ≈ 1o ↔ ( + = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∧ = {⟨⟨𝑍, 𝑍⟩, 𝑍⟩})))
 
Theoremrng1nfld 20558 The zero ring is not a field. (Contributed by AV, 29-Apr-2019.)
𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}       (𝑍𝑉𝑀 ∉ Field)
 
10.7.5  Left regular elements. More kinds of rings
 
Syntaxcrlreg 20559 Set of left-regular elements in a ring.
class RLReg
 
Syntaxcdomn 20560 Class of (ring theoretic) domains.
class Domn
 
Syntaxcidom 20561 Class of integral domains.
class IDomn
 
Syntaxcpid 20562 Class of principal ideal domains.
class PID
 
Definitiondf-rlreg 20563* 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(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
 
Definitiondf-domn 20564* 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][(0g𝑟) / 𝑧]𝑥𝑏𝑦𝑏 ((𝑥(.r𝑟)𝑦) = 𝑧 → (𝑥 = 𝑧𝑦 = 𝑧))}
 
Definitiondf-idom 20565 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
IDomn = (CRing ∩ Domn)
 
Definitiondf-pid 20566 A principal ideal domain is an integral domain satisfying the left principal ideal property. (Contributed by Stefan O'Rear, 29-Mar-2015.)
PID = (IDomn ∩ LPIR)
 
Theoremrrgval 20567* 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)    &    · = (.r𝑅)    &    0 = (0g𝑅)       𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
 
Theoremisrrg 20568* Membership in the set of left-regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
 
Theoremrrgeq0i 20569 Property of a left-regular element. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
 
Theoremrrgeq0 20570 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)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐸𝑌𝐵) → ((𝑋 · 𝑌) = 0𝑌 = 0 ))
 
Theoremrrgsupp 20571 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)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑋𝐸)    &   (𝜑𝑌:𝐼𝐵)       (𝜑 → (((𝐼 × {𝑋}) ∘f · 𝑌) supp 0 ) = (𝑌 supp 0 ))
 
Theoremrrgss 20572 Left-regular elements are a subset of the base set. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝐵 = (Base‘𝑅)       𝐸𝐵
 
Theoremunitrrg 20573 Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
𝐸 = (RLReg‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑅 ∈ Ring → 𝑈𝐸)
 
Theoremisdomn 20574* Expand definition of a domain. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 0 → (𝑥 = 0𝑦 = 0 ))))
 
Theoremdomnnzr 20575 A domain is a nonzero ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ NzRing)
 
Theoremdomnring 20576 A domain is a ring. (Contributed by Mario Carneiro, 28-Mar-2015.)
(𝑅 ∈ Domn → 𝑅 ∈ Ring)
 
Theoremdomneq0 20577 In a domain, a product is zero iff it has a zero factor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0𝑌 = 0 )))
 
Theoremdomnmuln0 20578 In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ (𝑋𝐵𝑋0 ) ∧ (𝑌𝐵𝑌0 )) → (𝑋 · 𝑌) ≠ 0 )
 
Theoremisdomn2 20579 A ring is a domain iff all nonzero elements are nonzero-divisors. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ (𝐵 ∖ { 0 }) ⊆ 𝐸))
 
Theoremdomnrrg 20580 In a domain, any nonzero element is a nonzero-divisor. (Contributed by Mario Carneiro, 28-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝐸 = (RLReg‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Domn ∧ 𝑋𝐵𝑋0 ) → 𝑋𝐸)
 
Theoremopprdomn 20581 The opposite of a domain is also a domain. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝑂 = (oppr𝑅)       (𝑅 ∈ Domn → 𝑂 ∈ Domn)
 
Theoremabvn0b 20582 Another characterization of domains, hinted at in abvtriv 20110: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.)
𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅))
 
Theoremdrngdomn 20583 A division ring is a domain. (Contributed by Mario Carneiro, 29-Mar-2015.)
(𝑅 ∈ DivRing → 𝑅 ∈ Domn)
 
Theoremisidom 20584 An integral domain is a commutative domain. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ IDomn ↔ (𝑅 ∈ CRing ∧ 𝑅 ∈ Domn))
 
Theoremfldidom 20585 A field is an integral domain. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof shortened by SN, 11-Nov-2024.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)
 
TheoremfldidomOLD 20586 Obsolete version of fldidom 20585 as of 11-Nov-2024. (Contributed by Mario Carneiro, 29-Mar-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑅 ∈ Field → 𝑅 ∈ IDomn)
 
Theoremfidomndrnglem 20587* Lemma for fidomndrng 20588. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &    = (∥r𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Domn)    &   (𝜑𝐵 ∈ Fin)    &   (𝜑𝐴 ∈ (𝐵 ∖ { 0 }))    &   𝐹 = (𝑥𝐵 ↦ (𝑥 · 𝐴))       (𝜑𝐴 1 )
 
Theoremfidomndrng 20588 A finite domain is a division ring. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ Domn ↔ 𝑅 ∈ DivRing))
 
Theoremfiidomfld 20589 A finite integral domain is a field. (Contributed by Mario Carneiro, 15-Jun-2015.)
𝐵 = (Base‘𝑅)       (𝐵 ∈ Fin → (𝑅 ∈ IDomn ↔ 𝑅 ∈ Field))
 
10.8  The complex numbers as an algebraic extensible structure
 
10.8.1  Definition and basic properties
 
Syntaxcpsmet 20590 Extend class notation with the class of all pseudometric spaces.
class PsMet
 
Syntaxcxmet 20591 Extend class notation with the class of all extended metric spaces.
class ∞Met
 
Syntaxcmet 20592 Extend class notation with the class of all metrics.
class Met
 
Syntaxcbl 20593 Extend class notation with the metric space ball function.
class ball
 
Syntaxcfbas 20594 Extend class definition to include the class of filter bases.
class fBas
 
Syntaxcfg 20595 Extend class definition to include the filter generating function.
class filGen
 
Syntaxcmopn 20596 Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
 
Syntaxcmetu 20597 Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
 
Definitiondf-psmet 20598* Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
PsMet = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥 ((𝑦𝑑𝑦) = 0 ∧ ∀𝑧𝑥𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
 
Definitiondf-xmet 20599* Define the set of all extended metrics on a given base set. The definition is similar to df-met 20600, but we also allow the metric to take on the value +∞. (Contributed by Mario Carneiro, 20-Aug-2015.)
∞Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ*m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) +𝑒 (𝑤𝑑𝑧)))})
 
Definitiondf-met 20600* Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric; see df-ms 23483. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. The 4 properties in Gleason's definition are shown by met0 23505, metgt0 23521, metsym 23512, and mettri 23514. (Contributed by NM, 25-Aug-2006.)
Met = (𝑥 ∈ V ↦ {𝑑 ∈ (ℝ ↑m (𝑥 × 𝑥)) ∣ ∀𝑦𝑥𝑧𝑥 (((𝑦𝑑𝑧) = 0 ↔ 𝑦 = 𝑧) ∧ ∀𝑤𝑥 (𝑦𝑑𝑧) ≤ ((𝑤𝑑𝑦) + (𝑤𝑑𝑧)))})
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