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Theorem List for Metamath Proof Explorer - 18701-18800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdvdsr01 18701 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 19331.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → 𝑋 0 )

Theoremdvdsr02 18702 Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
𝐵 = (Base‘𝑅)    &    = (∥r𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → ( 0 𝑋𝑋 = 0 ))

Theoremisunit 18703 Property of being a unit of a ring. A unit is an element that left- and right-divides one. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 8-Dec-2015.)
𝑈 = (Unit‘𝑅)    &    1 = (1r𝑅)    &    = (∥r𝑅)    &   𝑆 = (oppr𝑅)    &   𝐸 = (∥r𝑆)       (𝑋𝑈 ↔ (𝑋 1𝑋𝐸 1 ))

Theorem1unit 18704 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝑈 = (Unit‘𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 1𝑈)

Theoremunitcl 18705 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑋𝑈𝑋𝐵)

Theoremunitss 18706 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐵 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)       𝑈𝐵

Theoremopprunit 18707 Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝑆 = (oppr𝑅)       𝑈 = (Unit‘𝑆)

Theoremcrngunit 18708 Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝑈 = (Unit‘𝑅)    &    1 = (1r𝑅)    &    = (∥r𝑅)       (𝑅 ∈ CRing → (𝑋𝑈𝑋 1 ))

Theoremdvdsunit 18709 A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝑈 = (Unit‘𝑅)    &    = (∥r𝑅)       ((𝑅 ∈ CRing ∧ 𝑌 𝑋𝑋𝑈) → 𝑌𝑈)

Theoremunitmulcl 18710 The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝑈) → (𝑋 · 𝑌) ∈ 𝑈)

Theoremunitmulclb 18711 Reversal of unitmulcl 18710 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
𝑈 = (Unit‘𝑅)    &    · = (.r𝑅)    &   𝐵 = (Base‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → ((𝑋 · 𝑌) ∈ 𝑈 ↔ (𝑋𝑈𝑌𝑈)))

Theoremunitgrpbas 18712 The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       𝑈 = (Base‘𝐺)

Theoremunitgrp 18713 The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       (𝑅 ∈ Ring → 𝐺 ∈ Grp)

Theoremunitabl 18714 The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       (𝑅 ∈ CRing → 𝐺 ∈ Abel)

Theoremunitgrpid 18715 The identity of the multiplicative group is 1r. (Contributed by Mario Carneiro, 2-Dec-2014.)
𝑈 = (Unit‘𝑅)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → 1 = (0g𝐺))

Theoremunitsubm 18716 The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (Unit‘𝑅)    &   𝑀 = (mulGrp‘𝑅)       (𝑅 ∈ Ring → 𝑈 ∈ (SubMnd‘𝑀))

Syntaxcinvr 18717 Extend class notation with multiplicative inverse.
class invr

Definitiondf-invr 18718 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))))

Theoreminvrfval 18719 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 18720 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 18721 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 18722 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 18723 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 18724 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 18725 The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.)
𝐼 = (invr𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ Ring → (𝐼1 ) = 1 )

Theorem0unit 18726 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 18727 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 18728 Extend class notation with ring division.
class /r

Definitiondf-dvr 18729* Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
/r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))

Theoremdvrfval 18730* 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 18731 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 18732 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 18733 The units are closed under division. (Contributed by Mario Carneiro, 2-Jul-2014.)
𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈𝑌𝑈) → (𝑋 / 𝑌) ∈ 𝑈)

Theoremdvrid 18734 A cancellation law for division. (divid 10752 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝑈) → (𝑋 / 𝑋) = 1 )

Theoremdvr1 18735 A cancellation law for division. (div1 10754 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
𝐵 = (Base‘𝑅)    &    / = (/r𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐵) → (𝑋 / 1 ) = 𝑋)

Theoremdvrass 18736 An associative law for division. (divass 10741 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 18737 A cancellation law for division. (divcan1 10732 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 18738 A cancellation law for division. (divcan3 10749 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 18739 A cancellation law for division. (diveq1 10756 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 18740 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 18741* 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 18742* 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 18743* 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 18744* 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 18745* 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 18746* 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 18747* Expand out the class difference from isirred 18745. (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 18748 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 18749 The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐼) → 𝑋0 )

Theoremirredcl 18750 An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝐵 = (Base‘𝑅)       (𝑋𝐼𝑋𝐵)

Theoremirrednu 18751 An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &   𝑈 = (Unit‘𝑅)       (𝑋𝐼 → ¬ 𝑋𝑈)

Theoremirredn1 18752 The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝐼 = (Irred‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋𝐼) → 𝑋1 )

Theoremirredrmul 18753 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 18754 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 18755 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 18756 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 18757 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 primes

Syntaxcrpm 18758 Syntax for the ring primes function.
class RPrime

Definitiondf-rprm 18759* Define the function associating with a ring its set of prime elements. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 15472. Prime elements are closely related to irreducible elements ( see df-irred 18689). (Contributed by Mario Carneiro, 17-Feb-2015.)
RPrime = (𝑤 ∈ V ↦ (Base‘𝑤) / 𝑏{𝑝 ∈ (𝑏 ∖ ((Unit‘𝑤) ∪ {(0g𝑤)})) ∣ ∀𝑥𝑏𝑦𝑏 [(∥r𝑤) / 𝑑](𝑝𝑑(𝑥(.r𝑤)𝑦) → (𝑝𝑑𝑥𝑝𝑑𝑦))})

10.4.7  Ring homomorphisms

Syntaxcrh 18760 Extend class notation with the ring homomorphisms.
class RingHom

Syntaxcrs 18761 Extend class notation with the ring isomorphisms.
class RingIso

Syntaxcric 18762 Extend class notation with the ring isomorphism relation.
class 𝑟

Definitiondf-rnghom 18763* 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 18764* 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 18765* 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 18766 Define the ring isomorphism relation, analogous to df-gic 17749: 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 18767 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)

Theoremrhmrcl2 18768 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)

Theoremisrhm 18769 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 18770 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 18771 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 18772 Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))

Theoremrhmghm 18773 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 18774 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 18775 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 18776* 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 18777* 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 18778 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 18779 The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅))

Theoremrhmf1o 18780 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 18781 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 18782 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 18783 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 18784 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 18785 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 18786* 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 18787* 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 18788 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 18789 Alternate proof for f1rhm0to0 18788. Using ghmf1 17736 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 18790 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 18791 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 18792 The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
(𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)

Theorembrric2 18793* The relation "is isomorphic to" for (unital) rings. This theorem corresponds to the definition df-risc 33912 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 18794 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 18795 Extend class notation with class of all division rings.
class DivRing

Syntaxcfield 18796 Class of fields.
class Field

Definitiondf-drng 18797 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 18798 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field = (DivRing ∩ CRing)

Theoremisdrng 18799 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 18800 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 )))

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