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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcinvr 18401 Extend class notation with multiplicative inverse.
class invr

Definitiondf-invr 18402 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))

Theoreminvrfval 18403 Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    &   𝐼 = (invr𝑅)       𝐼 = (invg𝐺)

Theoremunitinvcl 18404 The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝐼𝑋) ∈ 𝑈)

Theoremunitinvinv 18405 The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝐼‘(𝐼𝑋)) = 𝑋)

Theoremringinvcl 18406 The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝐼𝑋) ∈ 𝐵)

Theoremunitlinv 18407 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → ((𝐼𝑋) · 𝑋) = 1 )

Theoremunitrinv 18408 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋 · (𝐼𝑋)) = 1 )

Theorem1rinv 18409 The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝐼 = (invr𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐼1 ) = 1 )

Theorem0unit 18410 The additive identity is a unit if and only if 1 = 0, i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → ( 0𝑈1 = 0 ))

Theoremunitnegcl 18411 The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑁𝑋) ∈ 𝑈)

Syntaxcdvr 18412 Extend class notation with ring division.
class /r

Definitiondf-dvr 18413* Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
/r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))

Theoremdvrfval 18414* Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &    / = (/r𝑅)        / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))

Theoremdvrval 18415 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &    / = (/r𝑅)       ((𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼𝑌)))

Theoremdvrcl 18416 Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) ∈ 𝐵)

Theoremunitdvcl 18417 The units are closed under division. (Contributed by Mario Carneiro, 2-Jul-2014.)
𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝑈) → (𝑋 / 𝑌) ∈ 𝑈)

Theoremdvrid 18418 A cancellation law for division. (divid 10463 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋 / 𝑋) = 1 )

Theoremdvr1 18419 A cancellation law for division. (div1 10465 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
𝐵 = (Base‘𝑅)    &    / = (/r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋 / 1 ) = 𝑋)

Theoremdvrass 18420 An associative law for division. (divass 10452 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵𝑍𝑈)) → ((𝑋 · 𝑌) / 𝑍) = (𝑋 · (𝑌 / 𝑍)))

Theoremdvrcan1 18421 A cancellation law for division. (divcan1 10443 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝑈) → ((𝑋 / 𝑌) · 𝑌) = 𝑋)

Theoremdvrcan3 18422 A cancellation law for division. (divcan3 10460 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝑈) → ((𝑋 · 𝑌) / 𝑌) = 𝑋)

Theoremdvreq1 18423 A cancellation law for division. (diveq1 10467 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵𝑌𝑈) → ((𝑋 / 𝑌) = 1𝑋 = 𝑌))

Theoremringinvdv 18424 Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝐼𝑋) = ( 1 / 𝑋))

Theoremrngidpropd 18425* The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (1r𝐾) = (1r𝐿))

Theoremdvdsrpropd 18426* The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (∥r𝐾) = (∥r𝐿))

Theoremunitpropd 18427* The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (Unit‘𝐾) = (Unit‘𝐿))

Theoreminvrpropd 18428* The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (invr𝐾) = (invr𝐿))

Theoremisirred 18429* An irreducible element of a ring is a non-unit that is not the product of two non-units. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (Irred‘𝑅)    &   𝑁 = (𝐵𝑈)    &    · = (.r𝑅)       (𝑋𝐼 ↔ (𝑋𝑁 ∧ ∀𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) ≠ 𝑋))

Theoremisnirred 18430* The property of being a non-irreducible (reducible) element in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (Irred‘𝑅)    &   𝑁 = (𝐵𝑈)    &    · = (.r𝑅)       (𝑋𝐵 → (¬ 𝑋𝐼 ↔ (𝑋𝑈 ∨ ∃𝑥𝑁𝑦𝑁 (𝑥 · 𝑦) = 𝑋)))

Theoremisirred2 18431* Expand out the class difference from isirred 18429. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (Irred‘𝑅)    &    · = (.r𝑅)       (𝑋𝐼 ↔ (𝑋𝐵 ∧ ¬ 𝑋𝑈 ∧ ∀𝑥𝐵𝑦𝐵 ((𝑥 · 𝑦) = 𝑋 → (𝑥𝑈𝑦𝑈))))

Theoremopprirred 18432 Irreducibility is symmetric, so the irreducible elements of the opposite ring are the same as the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (oppr𝑅)    &   𝐼 = (Irred‘𝑅)       𝐼 = (Irred‘𝑆)

Theoremirredn0 18433 The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐼) → 𝑋0 )

Theoremirredcl 18434 An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑋𝐼𝑋𝐵)

Theoremirrednu 18435 An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑋𝐼 → ¬ 𝑋𝑈)

Theoremirredn1 18436 The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐼) → 𝑋1 )

Theoremirredrmul 18437 The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐼𝑌𝑈) → (𝑋 · 𝑌) ∈ 𝐼)

Theoremirredlmul 18438 The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝐼) → (𝑋 · 𝑌) ∈ 𝐼)

Theoremirredmul 18439 If product of two elements is irreducible, then one of the elements must be a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)       ((𝑋𝐵𝑌𝐵 ∧ (𝑋 · 𝑌) ∈ 𝐼) → (𝑋𝑈𝑌𝑈))

Theoremirredneg 18440 The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝑁 = (invg𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐼) → (𝑁𝑋) ∈ 𝐼)

Theoremirrednegb 18441 An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝑁 = (invg𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋𝐼 ↔ (𝑁𝑋) ∈ 𝐼))

10.4.6  Ring homomorphisms

Syntaxcrh 18442 Extend class notation with the ring homomorphisms.
class RingHom

Syntaxcrs 18443 Extend class notation with the ring isomorphisms.
class RingIso

Syntaxcric 18444 Extend class notation with the ring isomorphism relation.
class 𝑟

Definitiondf-rnghom 18445* Define the set of ring homomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ (Base‘𝑟) / 𝑣(Base‘𝑠) / 𝑤{𝑓 ∈ (𝑤𝑚 𝑣) ∣ ((𝑓‘(1r𝑟)) = (1r𝑠) ∧ ∀𝑥𝑣𝑦𝑣 ((𝑓‘(𝑥(+g𝑟)𝑦)) = ((𝑓𝑥)(+g𝑠)(𝑓𝑦)) ∧ (𝑓‘(𝑥(.r𝑟)𝑦)) = ((𝑓𝑥)(.r𝑠)(𝑓𝑦))))})

Definitiondf-rngiso 18446* Define the set of ring isomorphisms from 𝑟 to 𝑠. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingIso = (𝑟 ∈ V, 𝑠 ∈ V ↦ {𝑓 ∈ (𝑟 RingHom 𝑠) ∣ 𝑓 ∈ (𝑠 RingHom 𝑟)})

Theoremdfrhm2 18447* The property of a ring homomorphism can be decomposed into separate homomorphic conditions for addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
RingHom = (𝑟 ∈ Ring, 𝑠 ∈ Ring ↦ ((𝑟 GrpHom 𝑠) ∩ ((mulGrp‘𝑟) MndHom (mulGrp‘𝑠))))

Definitiondf-ric 18448 Define the ring isomorphism relation, analogous to df-gic 17417: Two (unital) rings are said to be isomorphic iff they are connected by at least one isomorphism. Isomorphic rings share all global ring properties, but to relate local properties requires knowledge of a specific isomorphism. (Contributed by AV, 24-Dec-2019.)
𝑟 = ( RingIso “ (V ∖ 1𝑜))

Theoremrhmrcl1 18449 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)

Theoremrhmrcl2 18450 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)

Theoremisrhm 18451 A function is a ring homomorphism iff it preserves both addition and multiplication. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑁 = (mulGrp‘𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ (𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹 ∈ (𝑀 MndHom 𝑁))))

Theoremrhmmhm 18452 A ring homomorphism is a homomorphism of multiplicative monoids. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑀 = (mulGrp‘𝑅)    &   𝑁 = (mulGrp‘𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑀 MndHom 𝑁))

Theoremisrim0 18453 An isomorphism of rings is a homomorphism whose converse is also a homomorphism . (Contributed by AV, 22-Oct-2019.)
((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹 ∈ (𝑆 RingHom 𝑅))))

Theoremrimrcl 18454 Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Theoremrhmghm 18455 A ring homomorphism is an additive group homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Theoremrhmf 18456 A ring homomorphism is a function. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → 𝐹:𝐵𝐶)

Theoremrhmmul 18457 A homomorphism of rings preserves multiplication. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑋 = (Base‘𝑅)    &    · = (.r𝑅)    &    × = (.r𝑆)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐴𝑋𝐵𝑋) → (𝐹‘(𝐴 · 𝐵)) = ((𝐹𝐴) × (𝐹𝐵)))

Theoremisrhm2d 18458* Demonstration of ring homomorphism. (Contributed by Mario Carneiro, 13-Jun-2015.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &   𝑁 = (1r𝑆)    &    · = (.r𝑅)    &    × = (.r𝑆)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑 → (𝐹1 ) = 𝑁)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))    &   (𝜑𝐹 ∈ (𝑅 GrpHom 𝑆))       (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))

Theoremisrhmd 18459* Demonstration of ring homomorphism. (Contributed by Stefan O'Rear, 8-Mar-2015.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &   𝑁 = (1r𝑆)    &    · = (.r𝑅)    &    × = (.r𝑆)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑 → (𝐹1 ) = 𝑁)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 · 𝑦)) = ((𝐹𝑥) × (𝐹𝑦)))    &   𝐶 = (Base‘𝑆)    &    + = (+g𝑅)    &    = (+g𝑆)    &   (𝜑𝐹:𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))       (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))

Theoremrhm1 18460 Ring homomorphisms are required to fix 1. (Contributed by Stefan O'Rear, 8-Mar-2015.)
1 = (1r𝑅)    &   𝑁 = (1r𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹1 ) = 𝑁)

Theoremidrhm 18461 The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅))

Theoremrhmf1o 18462 A ring homomorphism is bijective iff its converse is also a ring homomorphism. (Contributed by AV, 22-Oct-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐵1-1-onto𝐶𝐹 ∈ (𝑆 RingHom 𝑅)))

Theoremisrim 18463 An isomorphism of rings is a bijective homomorphism. (Contributed by AV, 22-Oct-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       ((𝑅𝑉𝑆𝑊) → (𝐹 ∈ (𝑅 RingIso 𝑆) ↔ (𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶)))

Theoremrimf1o 18464 An isomorphism of rings is a bijection. (Contributed by AV, 22-Oct-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹:𝐵1-1-onto𝐶)

Theoremrimrhm 18465 An isomorphism of rings is a homomorphism. (Contributed by AV, 22-Oct-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 RingHom 𝑆))

Theoremrimgim 18466 An isomorphism of rings is an isomorphism of their additive groups. (Contributed by AV, 24-Dec-2019.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))

Theoremrhmco 18467 The composition of ring homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
((𝐹 ∈ (𝑇 RingHom 𝑈) ∧ 𝐺 ∈ (𝑆 RingHom 𝑇)) → (𝐹𝐺) ∈ (𝑆 RingHom 𝑈))

Theorempwsco1rhm 18468* Right composition with a function on the index sets yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑌 = (𝑅s 𝐴)    &   𝑍 = (𝑅s 𝐵)    &   𝐶 = (Base‘𝑍)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑𝐹:𝐴𝐵)       (𝜑 → (𝑔𝐶 ↦ (𝑔𝐹)) ∈ (𝑍 RingHom 𝑌))

Theorempwsco2rhm 18469* Left composition with a ring homomorphism yields a ring homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
𝑌 = (𝑅s 𝐴)    &   𝑍 = (𝑆s 𝐴)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝐴𝑉)    &   (𝜑𝐹 ∈ (𝑅 RingHom 𝑆))       (𝜑 → (𝑔𝐵 ↦ (𝐹𝑔)) ∈ (𝑌 RingHom 𝑍))

Theoremf1rhm0to0 18470 If a ring homomorphism 𝐹 is injective, it maps the zero of one ring (and only the zero) to the zero of the other ring. (Contributed by AV, 24-Oct-2019.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑆)    &    0 = (0g𝑅)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))

Theoremf1rhm0to0ALT 18471 Alternate proof for f1rhm0to0 18470. Using ghmf1 17404 does not make the proof shorter and requires disjoint variable restrictions! (Contributed by AV, 24-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑆)    &    0 = (0g𝑅)       ((𝐹 ∈ (𝑅 RingHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))

Theoremrim0to0 18472 A ring isomorphism maps the zero of one ring (and only the zero) to the zero of the other ring. (Contributed by AV, 24-Oct-2019.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑆)    &    0 = (0g𝑅)       ((𝐹 ∈ (𝑅 RingIso 𝑆) ∧ 𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))

Theoremkerf1hrm 18473 A ring homomorphism 𝐹 is injective if and only if its kernel is the singleton {𝑁}. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑅)    &    0 = (0g𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ (𝐹 “ { 0 }) = {𝑁}))

Theorembrric 18474 The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
(𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Theorembrric2 18475* The relation "is isomorphic to" for (unital) rings. This theorem corresponds to the definition df-risc 32842 of the ring isomorphism relation in JM's mathbox. (Contributed by AV, 24-Dec-2019.)
(𝑅𝑟 𝑆 ↔ ((𝑅 ∈ Ring ∧ 𝑆 ∈ Ring) ∧ ∃𝑓 𝑓 ∈ (𝑅 RingIso 𝑆)))

Theoremricgic 18476 If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.)
(𝑅𝑟 𝑆𝑅𝑔 𝑆)

10.5  Division rings and fields

10.5.1  Definition and basic properties

Syntaxcdr 18477 Extend class notation with class of all division rings.
class DivRing

Syntaxcfield 18478 Class of fields.
class Field

Definitiondf-drng 18479 Define class of all division rings. A division ring is a ring in which the set of units is exactly the nonzero elements of the ring. (Contributed by NM, 18-Oct-2012.)
DivRing = {𝑟 ∈ Ring ∣ (Unit‘𝑟) = ((Base‘𝑟) ∖ {(0g𝑟)})}

Definitiondf-field 18480 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field = (DivRing ∩ CRing)

Theoremisdrng 18481 The predicate "is a division ring". (Contributed by NM, 18-Oct-2012.) (Revised by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝑈 = (𝐵 ∖ { 0 })))

Theoremdrngunit 18482 Elementhood in the set of units when 𝑅 is a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)       (𝑅 ∈ DivRing → (𝑋𝑈 ↔ (𝑋𝐵𝑋0 )))

Theoremdrngui 18483 The set of units of a division ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑅 ∈ DivRing       (𝐵 ∖ { 0 }) = (Unit‘𝑅)

Theoremdrngring 18484 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing → 𝑅 ∈ Ring)

Theoremdrnggrp 18485 A division ring is a group. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing → 𝑅 ∈ Grp)

Theoremisfld 18486 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))

Theoremisdrng2 18487 A division ring can equivalently be defined as a ring such that the nonzero elements form a group under multiplication (from which it follows that this is the same group as the group of units). (Contributed by Mario Carneiro, 2-Dec-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))       (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ 𝐺 ∈ Grp))

Theoremdrngprop 18488 If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
(Base‘𝐾) = (Base‘𝐿)    &   (+g𝐾) = (+g𝐿)    &   (.r𝐾) = (.r𝐿)       (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)

Theoremdrngmgp 18489 A division ring contains a multiplicative group. (Contributed by NM, 8-Sep-2011.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))       (𝑅 ∈ DivRing → 𝐺 ∈ Grp)

Theoremdrngmcl 18490 The product of two nonzero elements of a division ring is nonzero. (Contributed by NM, 7-Sep-2011.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋 ∈ (𝐵 ∖ { 0 }) ∧ 𝑌 ∈ (𝐵 ∖ { 0 })) → (𝑋 · 𝑌) ∈ (𝐵 ∖ { 0 }))

Theoremdrngid 18491 A division ring's unit is the identity element of its multiplicative group. (Contributed by NM, 7-Sep-2011.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s (𝐵 ∖ { 0 }))       (𝑅 ∈ DivRing → 1 = (0g𝐺))

Theoremdrngunz 18492 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)
0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ DivRing → 10 )

Theoremdrngid2 18493 Properties showing that an element 𝐼 is the identity element of a division ring. (Contributed by Mario Carneiro, 11-Oct-2013.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &    0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ DivRing → ((𝐼𝐵𝐼0 ∧ (𝐼 · 𝐼) = 𝐼) ↔ 1 = 𝐼))

Theoremdrnginvrcl 18494 Closure of the multiplicative inverse in a division ring. (reccl 10441 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝐼𝑋) ∈ 𝐵)

Theoremdrnginvrn0 18495 The multiplicative inverse in a division ring is nonzero. (recne0 10447 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝐼𝑋) ≠ 0 )

Theoremdrnginvrl 18496 Property of the multiplicative inverse in a division ring. (recid2 10449 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → ((𝐼𝑋) · 𝑋) = 1 )

Theoremdrnginvrr 18497 Property of the multiplicative inverse in a division ring. (recid 10448 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝑋 · (𝐼𝑋)) = 1 )

Theoremdrngmul0or 18498 A product is zero iff one of its factors is zero. (Contributed by NM, 8-Oct-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0𝑌 = 0 )))

Theoremdrngmulne0 18499 A product is nonzero iff both its factors are nonzero. (Contributed by NM, 18-Oct-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝑋 · 𝑌) ≠ 0 ↔ (𝑋0𝑌0 )))

Theoremdrngmuleq0 18500 An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ DivRing)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)    &   (𝜑𝑌0 )       (𝜑 → ((𝑋 · 𝑌) = 0𝑋 = 0 ))

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