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Theorem List for Metamath Proof Explorer - 33701-33800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrngmgmbs4 33701* The range of an internal operation with a left and right identity element equals its base set. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
((𝐺:(𝑋 × 𝑋)⟶𝑋 ∧ ∃𝑢𝑋𝑥𝑋 ((𝑢𝐺𝑥) = 𝑥 ∧ (𝑥𝐺𝑢) = 𝑥)) → ran 𝐺 = 𝑋)

Theoremrngodm1dm2 33702 In a unital ring the domain of the first variable of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
𝐻 = (2nd𝑅)    &   𝐺 = (1st𝑅)       (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)

Theoremrngorn1 33703 In a unital ring the range of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
𝐻 = (2nd𝑅)    &   𝐺 = (1st𝑅)       (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻)

Theoremrngorn1eq 33704 In a unital ring the range of the addition equals the range of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)
𝐻 = (2nd𝑅)    &   𝐺 = (1st𝑅)       (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻)

Theoremrngomndo 33705 In a unital ring the multiplication is a monoid. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
𝐻 = (2nd𝑅)       (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Theoremrngoidmlem 33706 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
𝐻 = (2nd𝑅)    &   𝑋 = ran (1st𝑅)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

Theoremrngolidm 33707 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
𝐻 = (2nd𝑅)    &   𝑋 = ran (1st𝑅)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑈𝐻𝐴) = 𝐴)

Theoremrngoridm 33708 The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)
𝐻 = (2nd𝑅)    &   𝑋 = ran (1st𝑅)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝐴𝐻𝑈) = 𝐴)

Theoremrngo1cl 33709 The unit of a ring belongs to the base set. (Contributed by FL, 12-Feb-2010.) (New usage is discouraged.)
𝑋 = ran (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑈 = (GId‘𝐻)       (𝑅 ∈ RingOps → 𝑈𝑋)

Theoremrngoueqz 33710 Obsolete as of 23-Jan-2020. Use 0ring01eqbi 19254 instead. In a unital ring the zero equals the unity iff the ring is the zero ring. (Contributed by FL, 14-Feb-2010.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → (𝑋 ≈ 1𝑜𝑈 = 𝑍))

Theoremrngonegmn1l 33711 Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = ((𝑁𝑈)𝐻𝐴))

Theoremrngonegmn1r 33712 Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐻(𝑁𝑈)))

Theoremrngoneglmul 33713 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = ((𝑁𝐴)𝐻𝐵))

Theoremrngonegrmul 33714 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑁 = (inv‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴𝐻𝐵)) = (𝐴𝐻(𝑁𝐵)))

Theoremrngosubdi 33715 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝐴𝐻(𝐵𝐷𝐶)) = ((𝐴𝐻𝐵)𝐷(𝐴𝐻𝐶)))

Theoremrngosubdir 33716 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝐷 = ( /𝑔𝐺)       ((𝑅 ∈ RingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐷(𝐵𝐻𝐶)))

Theoremzerdivemp1x 33717* In a unitary ring a left invertible element is not a zero divisor. See also ringinvnzdiv 18574. (Contributed by Jeff Madsen, 18-Apr-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐴𝑋 ∧ ∃𝑎𝑋 (𝑎𝐻𝐴) = 𝑈) → (𝐵𝑋 → ((𝐴𝐻𝐵) = 𝑍𝐵 = 𝑍)))

20.19.17  Division Rings

Syntaxcdrng 33718 Extend class notation with the class of all division rings.
class DivRingOps

Definitiondf-drngo 33719* Define the class of all division rings (sometimes called skew fields). A division ring is a unital ring where every element except the additive identity has a multiplicative inverse. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
DivRingOps = {⟨𝑔, ⟩ ∣ (⟨𝑔, ⟩ ∈ RingOps ∧ ( ↾ ((ran 𝑔 ∖ {(GId‘𝑔)}) × (ran 𝑔 ∖ {(GId‘𝑔)}))) ∈ GrpOp)}

Theoremisdivrngo 33720 The predicate "is a division ring". (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
(𝐻𝐴 → (⟨𝐺, 𝐻⟩ ∈ DivRingOps ↔ (⟨𝐺, 𝐻⟩ ∈ RingOps ∧ (𝐻 ↾ ((ran 𝐺 ∖ {(GId‘𝐺)}) × (ran 𝐺 ∖ {(GId‘𝐺)}))) ∈ GrpOp)))

Theoremdrngoi 33721 The properties of a division ring. (Contributed by NM, 4-Apr-2009.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ DivRingOps → (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Theoremgidsn 33722 Obsolete as of 23-Jan-2020. Use mnd1id 17313 instead. The identity element of the trivial group. (Contributed by FL, 21-Jun-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝐴 ∈ V       (GId‘{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}) = 𝐴

Theoremzrdivrng 33723 The zero ring is not a division ring. (Contributed by FL, 24-Jan-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝐴 ∈ V        ¬ ⟨{⟨⟨𝐴, 𝐴⟩, 𝐴⟩}, {⟨⟨𝐴, 𝐴⟩, 𝐴⟩}⟩ ∈ DivRingOps

Theoremdvrunz 33724 In a division ring the unit is different from the zero. (Contributed by FL, 14-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       (𝑅 ∈ DivRingOps → 𝑈𝑍)

Theoremisgrpda 33725* Properties that determine a group operation. (Contributed by Jeff Madsen, 1-Dec-2009.) (New usage is discouraged.)
(𝜑𝑋 ∈ V)    &   (𝜑𝐺:(𝑋 × 𝑋)⟶𝑋)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐺𝑦)𝐺𝑧) = (𝑥𝐺(𝑦𝐺𝑧)))    &   (𝜑𝑈𝑋)    &   ((𝜑𝑥𝑋) → (𝑈𝐺𝑥) = 𝑥)    &   ((𝜑𝑥𝑋) → ∃𝑛𝑋 (𝑛𝐺𝑥) = 𝑈)       (𝜑𝐺 ∈ GrpOp)

Theoremisdrngo1 33726 The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑋 = ran 𝐺       (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝐻 ↾ ((𝑋 ∖ {𝑍}) × (𝑋 ∖ {𝑍}))) ∈ GrpOp))

Theoremdivrngcl 33727 The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑋 = ran 𝐺       ((𝑅 ∈ DivRingOps ∧ 𝐴 ∈ (𝑋 ∖ {𝑍}) ∧ 𝐵 ∈ (𝑋 ∖ {𝑍})) → (𝐴𝐻𝐵) ∈ (𝑋 ∖ {𝑍}))

Theoremisdrngo2 33728* A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐻)       (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦 ∈ (𝑋 ∖ {𝑍})(𝑦𝐻𝑥) = 𝑈)))

Theoremisdrngo3 33729* A division ring is a ring in which 1 ≠ 0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑍 = (GId‘𝐺)    &   𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐻)       (𝑅 ∈ DivRingOps ↔ (𝑅 ∈ RingOps ∧ (𝑈𝑍 ∧ ∀𝑥 ∈ (𝑋 ∖ {𝑍})∃𝑦𝑋 (𝑦𝐻𝑥) = 𝑈)))

20.19.18  Ring homomorphisms

Syntaxcrnghom 33730 Extend class notation with the class of ring homomorphisms.
class RngHom

Syntaxcrngiso 33731 Extend class notation with the class of ring isomorphisms.
class RngIso

Syntaxcrisc 33732 Extend class notation with the ring isomorphism relation.
class 𝑟

Definitiondf-rngohom 33733* Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
RngHom = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (ran (1st𝑠) ↑𝑚 ran (1st𝑟)) ∣ ((𝑓‘(GId‘(2nd𝑟))) = (GId‘(2nd𝑠)) ∧ ∀𝑥 ∈ ran (1st𝑟)∀𝑦 ∈ ran (1st𝑟)((𝑓‘(𝑥(1st𝑟)𝑦)) = ((𝑓𝑥)(1st𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(2nd𝑟)𝑦)) = ((𝑓𝑥)(2nd𝑠)(𝑓𝑦))))})

Theoremrngohomval 33734* The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐻)    &   𝐽 = (1st𝑆)    &   𝐾 = (2nd𝑆)    &   𝑌 = ran 𝐽    &   𝑉 = (GId‘𝐾)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngHom 𝑆) = {𝑓 ∈ (𝑌𝑚 𝑋) ∣ ((𝑓𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝑓‘(𝑥𝐺𝑦)) = ((𝑓𝑥)𝐽(𝑓𝑦)) ∧ (𝑓‘(𝑥𝐻𝑦)) = ((𝑓𝑥)𝐾(𝑓𝑦))))})

Theoremisrngohom 33735* The predicate "is a ring homomorphism from 𝑅 to 𝑆." (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐻)    &   𝐽 = (1st𝑆)    &   𝐾 = (2nd𝑆)    &   𝑌 = ran 𝐽    &   𝑉 = (GId‘𝐾)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngHom 𝑆) ↔ (𝐹:𝑋𝑌 ∧ (𝐹𝑈) = 𝑉 ∧ ∀𝑥𝑋𝑦𝑋 ((𝐹‘(𝑥𝐺𝑦)) = ((𝐹𝑥)𝐽(𝐹𝑦)) ∧ (𝐹‘(𝑥𝐻𝑦)) = ((𝐹𝑥)𝐾(𝐹𝑦))))))

Theoremrngohomf 33736 A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹:𝑋𝑌)

Theoremrngohomcl 33737 Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ 𝐴𝑋) → (𝐹𝐴) ∈ 𝑌)

Theoremrngohom1 33738 A ring homomorphism preserves 1. (Contributed by Jeff Madsen, 24-Jun-2011.)
𝐻 = (2nd𝑅)    &   𝑈 = (GId‘𝐻)    &   𝐾 = (2nd𝑆)    &   𝑉 = (GId‘𝐾)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑈) = 𝑉)

Theoremrngohomadd 33739 Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)       (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐺𝐵)) = ((𝐹𝐴)𝐽(𝐹𝐵)))

Theoremrngohommul 33740 Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐻 = (2nd𝑅)    &   𝐾 = (2nd𝑆)       (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹𝐴)𝐾(𝐹𝐵)))

Theoremrngogrphom 33741 A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
𝐺 = (1st𝑅)    &   𝐽 = (1st𝑆)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → 𝐹 ∈ (𝐺 GrpOpHom 𝐽))

Theoremrngohom0 33742 A ring homomorphism preserves 0. (Contributed by Jeff Madsen, 2-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)    &   𝐽 = (1st𝑆)    &   𝑊 = (GId‘𝐽)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹𝑍) = 𝑊)

Theoremrngohomsub 33743 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐻 = ( /𝑔𝐺)    &   𝐽 = (1st𝑆)    &   𝐾 = ( /𝑔𝐽)       (((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) ∧ (𝐴𝑋𝐵𝑋)) → (𝐹‘(𝐴𝐻𝐵)) = ((𝐹𝐴)𝐾(𝐹𝐵)))

Theoremrngohomco 33744 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
(((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐺 ∈ (𝑆 RngHom 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngHom 𝑇))

Theoremrngokerinj 33745 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝑊 = (GId‘𝐺)    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽    &   𝑍 = (GId‘𝐽)       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngHom 𝑆)) → (𝐹:𝑋1-1𝑌 ↔ (𝐹 “ {𝑍}) = {𝑊}))

Definitiondf-rngoiso 33746* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)
RngIso = (𝑟 ∈ RingOps, 𝑠 ∈ RingOps ↦ {𝑓 ∈ (𝑟 RngHom 𝑠) ∣ 𝑓:ran (1st𝑟)–1-1-onto→ran (1st𝑠)})

Theoremrngoisoval 33747* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅 RngIso 𝑆) = {𝑓 ∈ (𝑅 RngHom 𝑆) ∣ 𝑓:𝑋1-1-onto𝑌})

Theoremisrngoiso 33748 The predicate "is a ring isomorphism between 𝑅 and 𝑆." (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝐹 ∈ (𝑅 RngIso 𝑆) ↔ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋1-1-onto𝑌)))

Theoremrngoiso1o 33749 A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹:𝑋1-1-onto𝑌)

Theoremrngoisohom 33750 A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑅 RngHom 𝑆))

Theoremrngoisocnv 33751 The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝐹 ∈ (𝑆 RngIso 𝑅))

Theoremrngoisoco 33752 The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
(((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝑇 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngIso 𝑆) ∧ 𝐺 ∈ (𝑆 RngIso 𝑇))) → (𝐺𝐹) ∈ (𝑅 RngIso 𝑇))

Definitiondf-risc 33753* Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑟 = {⟨𝑟, 𝑠⟩ ∣ ((𝑟 ∈ RingOps ∧ 𝑠 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑟 RngIso 𝑠))}

Theoremisriscg 33754* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅𝐴𝑆𝐵) → (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆))))

Theoremisrisc 33755* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝑅 ∈ V    &   𝑆 ∈ V       (𝑅𝑟 𝑆 ↔ ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) ∧ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))

Theoremrisc 33756* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps) → (𝑅𝑟 𝑆 ↔ ∃𝑓 𝑓 ∈ (𝑅 RngIso 𝑆)))

Theoremrisci 33757 Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RngIso 𝑆)) → 𝑅𝑟 𝑆)

Theoremriscer 33758 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
𝑟 Er dom ≃𝑟

20.19.19  Commutative rings

Syntaxccm2 33759 Extend class notation with a class that adds commutativity to various flavors of rings.
class Com2

Definitiondf-com2 33760* A device to add commutativity to various sorts of rings. I use ran 𝑔 because I suppose 𝑔 has a neutral element and therefore is onto. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
Com2 = {⟨𝑔, ⟩ ∣ ∀𝑎 ∈ ran 𝑔𝑏 ∈ ran 𝑔(𝑎𝑏) = (𝑏𝑎)}

Syntaxcfld 33761 Extend class notation with the class of all fields.
class Fld

Definitiondf-fld 33762 Definition of a field. A field is a commutative division ring. (Contributed by FL, 6-Sep-2009.) (Revised by Jeff Madsen, 10-Jun-2010.) (New usage is discouraged.)
Fld = (DivRingOps ∩ Com2)

Syntaxccring 33763 Extend class notation with the class of commutative rings.
class CRingOps

Definitiondf-crngo 33764 Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
CRingOps = (RingOps ∩ Com2)

Theoremiscom2 33765* A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))

Theoremiscrngo 33766 The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
(𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ 𝑅 ∈ Com2))

Theoremiscrngo2 33767* The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ CRingOps ↔ (𝑅 ∈ RingOps ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐻𝑦) = (𝑦𝐻𝑥)))

Theoremiscringd 33768* Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
(𝜑𝐺 ∈ AbelOp)    &   (𝜑𝑋 = ran 𝐺)    &   (𝜑𝐻:(𝑋 × 𝑋)⟶𝑋)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋𝑧𝑋)) → (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)))    &   (𝜑𝑈𝑋)    &   ((𝜑𝑦𝑋) → (𝑦𝐻𝑈) = 𝑦)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐻𝑦) = (𝑦𝐻𝑥))       (𝜑 → ⟨𝐺, 𝐻⟩ ∈ CRingOps)

Theoremflddivrng 33769 A field is a division ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
(𝐾 ∈ Fld → 𝐾 ∈ DivRingOps)

Theoremcrngorngo 33770 A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ CRingOps → 𝑅 ∈ RingOps)

Theoremcrngocom 33771 The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐻𝐵) = (𝐵𝐻𝐴))

Theoremcrngm23 33772 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐻𝐵)𝐻𝐶) = ((𝐴𝐻𝐶)𝐻𝐵))

Theoremcrngm4 33773 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ CRingOps ∧ (𝐴𝑋𝐵𝑋) ∧ (𝐶𝑋𝐷𝑋)) → ((𝐴𝐻𝐵)𝐻(𝐶𝐻𝐷)) = ((𝐴𝐻𝐶)𝐻(𝐵𝐻𝐷)))

Theoremfldcrng 33774 A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
(𝐾 ∈ Fld → 𝐾 ∈ CRingOps)

Theoremisfld2 33775 The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝐾 ∈ Fld ↔ (𝐾 ∈ DivRingOps ∧ 𝐾 ∈ CRingOps))

Theoremcrngohomfo 33776 The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺    &   𝐽 = (1st𝑆)    &   𝑌 = ran 𝐽       (((𝑅 ∈ CRingOps ∧ 𝑆 ∈ RingOps) ∧ (𝐹 ∈ (𝑅 RngHom 𝑆) ∧ 𝐹:𝑋onto𝑌)) → 𝑆 ∈ CRingOps)

20.19.20  Ideals

Syntaxcidl 33777 Extend class notation with the class of ideals.
class Idl

Syntaxcpridl 33778 Extend class notation with the class of prime ideals.
class PrIdl

Syntaxcmaxidl 33779 Extend class notation with the class of maximal ideals.
class MaxIdl

Definitiondf-idl 33780* Define the class of (two-sided) ideals of a ring 𝑅. A subset of 𝑅 is an ideal if it contains 0, is closed under addition, and is closed under multiplication on either side by any element of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
Idl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ 𝒫 ran (1st𝑟) ∣ ((GId‘(1st𝑟)) ∈ 𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥(1st𝑟)𝑦) ∈ 𝑖 ∧ ∀𝑧 ∈ ran (1st𝑟)((𝑧(2nd𝑟)𝑥) ∈ 𝑖 ∧ (𝑥(2nd𝑟)𝑧) ∈ 𝑖)))})

Definitiondf-pridl 33781* Define the class of prime ideals of a ring 𝑅. A proper ideal 𝐼 of 𝑅 is prime if whenever 𝐴𝐵𝐼 for ideals 𝐴 and 𝐵, either 𝐴𝐼 or 𝐵𝐼. The more familiar definition using elements rather than ideals is equivalent provided 𝑅 is commutative; see ispridl2 33808 and ispridlc 33840. (Contributed by Jeff Madsen, 10-Jun-2010.)
PrIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑎 ∈ (Idl‘𝑟)∀𝑏 ∈ (Idl‘𝑟)(∀𝑥𝑎𝑦𝑏 (𝑥(2nd𝑟)𝑦) ∈ 𝑖 → (𝑎𝑖𝑏𝑖)))})

Definitiondf-maxidl 33782* Define the class of maximal ideals of a ring 𝑅. A proper ideal is called maximal if it is maximal with respect to inclusion among proper ideals. (Contributed by Jeff Madsen, 5-Jan-2011.)
MaxIdl = (𝑟 ∈ RingOps ↦ {𝑖 ∈ (Idl‘𝑟) ∣ (𝑖 ≠ ran (1st𝑟) ∧ ∀𝑗 ∈ (Idl‘𝑟)(𝑖𝑗 → (𝑗 = 𝑖𝑗 = ran (1st𝑟))))})

Theoremidlval 33783* The class of ideals of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → (Idl‘𝑅) = {𝑖 ∈ 𝒫 𝑋 ∣ (𝑍𝑖 ∧ ∀𝑥𝑖 (∀𝑦𝑖 (𝑥𝐺𝑦) ∈ 𝑖 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝑖 ∧ (𝑥𝐻𝑧) ∈ 𝑖)))})

Theoremisidl 33784* The predicate "is an ideal of the ring 𝑅." (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 ((𝑧𝐻𝑥) ∈ 𝐼 ∧ (𝑥𝐻𝑧) ∈ 𝐼)))))

Theoremisidlc 33785* The predicate "is an ideal of the commutative ring 𝑅." (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ CRingOps → (𝐼 ∈ (Idl‘𝑅) ↔ (𝐼𝑋𝑍𝐼 ∧ ∀𝑥𝐼 (∀𝑦𝐼 (𝑥𝐺𝑦) ∈ 𝐼 ∧ ∀𝑧𝑋 (𝑧𝐻𝑥) ∈ 𝐼))))

Theoremidlss 33786 An ideal of 𝑅 is a subset of 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝐼𝑋)

Theoremidlcl 33787 An element of an ideal is an element of the ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → 𝐴𝑋)

Theoremidl0cl 33788 An ideal contains 0. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)       ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → 𝑍𝐼)

Theoremidladdcl 33789 An ideal is closed under addition. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐺𝐵) ∈ 𝐼)

Theoremidllmulcl 33790 An ideal is closed under multiplication on the left. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐵𝐻𝐴) ∈ 𝐼)

Theoremidlrmulcl 33791 An ideal is closed under multiplication on the right. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝑋)) → (𝐴𝐻𝐵) ∈ 𝐼)

Theoremidlnegcl 33792 An ideal is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑁 = (inv‘𝐺)       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ 𝐴𝐼) → (𝑁𝐴) ∈ 𝐼)

Theoremidlsubcl 33793 An ideal is closed under subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐷 = ( /𝑔𝐺)       (((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) ∧ (𝐴𝐼𝐵𝐼)) → (𝐴𝐷𝐵) ∈ 𝐼)

Theoremrngoidl 33794 A ring 𝑅 is an 𝑅 ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑋 = ran 𝐺       (𝑅 ∈ RingOps → 𝑋 ∈ (Idl‘𝑅))

Theorem0idl 33795 The set containing only 0 is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅))

Theorem1idl 33796 Two ways of expressing the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑈 = (GId‘𝐻)       ((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅)) → (𝑈𝐼𝐼 = 𝑋))

Theorem0rngo 33797 In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)    &   𝑈 = (GId‘𝐻)       (𝑅 ∈ RingOps → (𝑍 = 𝑈𝑋 = {𝑍}))

Theoremdivrngidl 33798 The only ideals in a division ring are the zero ideal and the unit ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st𝑅)    &   𝐻 = (2nd𝑅)    &   𝑋 = ran 𝐺    &   𝑍 = (GId‘𝐺)       (𝑅 ∈ DivRingOps → (Idl‘𝑅) = {{𝑍}, 𝑋})

Theoremintidl 33799 The intersection of a nonempty collection of ideals is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐶 ≠ ∅ ∧ 𝐶 ⊆ (Idl‘𝑅)) → 𝐶 ∈ (Idl‘𝑅))

Theoreminidl 33800 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idl‘𝑅) ∧ 𝐽 ∈ (Idl‘𝑅)) → (𝐼𝐽) ∈ (Idl‘𝑅))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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