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Theorem List for Metamath Proof Explorer - 19401-19500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-ric 19401 Define the ring isomorphism relation, analogous to df-gic 18340: 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 19402 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑅 ∈ Ring)
 
Theoremrhmrcl2 19403 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
(𝐹 ∈ (𝑅 RingHom 𝑆) → 𝑆 ∈ Ring)
 
Theoremisrhm 19404 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 19405 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 19406 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 19407 Reverse closure for an isomorphism of rings. (Contributed by AV, 22-Oct-2019.)
(𝐹 ∈ (𝑅 RingIso 𝑆) → (𝑅 ∈ V ∧ 𝑆 ∈ V))
 
Theoremrhmghm 19408 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 19409 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 19410 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 19411* 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 19412* 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 19413 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 19414 The identity homomorphism on a ring. (Contributed by AV, 14-Feb-2020.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → ( I ↾ 𝐵) ∈ (𝑅 RingHom 𝑅))
 
Theoremrhmf1o 19415 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 19416 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 19417 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 19418 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 19419 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 19420 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 19421* 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 19422* 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))
 
Theoremf1ghm0to0 19423 If a group homomorphism 𝐹 is injective, it maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 13-May-2023.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑆)    &    0 = (0g𝑅)       ((𝐹 ∈ (𝑅 GrpHom 𝑆) ∧ 𝐹:𝐴1-1𝐵𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
 
Theoremf1rhm0to0OLD 19424 Obsolete version of f1ghm0to0 19423 as of 13-May-2023. (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 ))
 
Theoremf1rhm0to0ALT 19425 Alternate proof for f1ghm0to0 19423. Using ghmf1 18327 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 ))
 
Theoremgim0to0 19426 A group isomorphism maps the zero of one group (and only the zero) to the zero of the other group. (Contributed by AV, 24-Oct-2019.) (Revised by Thierry Arnoux, 23-May-2023.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑆)    &    0 = (0g𝑅)       ((𝐹 ∈ (𝑅 GrpIso 𝑆) ∧ 𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
 
Theoremrim0to0OLD 19427 Obsolete version of gim0to0 19426 as of 13-May-2023. (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 RingIso 𝑆) ∧ 𝑋𝐴) → ((𝐹𝑋) = 𝑁𝑋 = 0 ))
 
Theoremkerf1ghm 19428 A group 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.) (Revised by Thierry Arnoux, 13-May-2023.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑅)    &    0 = (0g𝑆)       (𝐹 ∈ (𝑅 GrpHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ (𝐹 “ { 0 }) = {𝑁}))
 
Theoremkerf1hrmOLD 19429 Obsolete version of kerf1ghm 19428 as of 13-May-2023. (Contributed by Thierry Arnoux, 27-Oct-2017.) (Proof shortened by AV, 24-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
𝐴 = (Base‘𝑅)    &   𝐵 = (Base‘𝑆)    &   𝑁 = (0g𝑅)    &    0 = (0g𝑆)       (𝐹 ∈ (𝑅 RingHom 𝑆) → (𝐹:𝐴1-1𝐵 ↔ (𝐹 “ { 0 }) = {𝑁}))
 
Theorembrric 19430 The relation "is isomorphic to" for (unital) rings. (Contributed by AV, 24-Dec-2019.)
(𝑅𝑟 𝑆 ↔ (𝑅 RingIso 𝑆) ≠ ∅)
 
Theorembrric2 19431* The relation "is isomorphic to" for (unital) rings. This theorem corresponds to the definition df-risc 35144 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 19432 If two rings are (ring) isomorphic, their additive groups are (group) isomorphic. (Contributed by AV, 24-Dec-2019.)
(𝑅𝑟 𝑆𝑅𝑔 𝑆)
 
10.4  Division rings and fields
 
10.4.1  Definition and basic properties
 
Syntaxcdr 19433 Extend class notation with class of all division rings.
class DivRing
 
Syntaxcfield 19434 Class of fields.
class Field
 
Definitiondf-drng 19435 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 19436 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
Field = (DivRing ∩ CRing)
 
Theoremisdrng 19437 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 19438 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 19439 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 19440 A division ring is a ring. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing → 𝑅 ∈ Ring)
 
Theoremdrnggrp 19441 A division ring is a group. (Contributed by NM, 8-Sep-2011.)
(𝑅 ∈ DivRing → 𝑅 ∈ Grp)
 
Theoremisfld 19442 A field is a commutative division ring. (Contributed by Mario Carneiro, 17-Jun-2015.)
(𝑅 ∈ Field ↔ (𝑅 ∈ DivRing ∧ 𝑅 ∈ CRing))
 
Theoremisdrng2 19443 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 19444 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 19445 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 19446 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 19447 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 19448 A division ring's unit is different from its zero. (Contributed by NM, 8-Sep-2011.)
0 = (0g𝑅)    &    1 = (1r𝑅)       (𝑅 ∈ DivRing → 10 )
 
Theoremdrngid2 19449 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 19450 Closure of the multiplicative inverse in a division ring. (reccl 11294 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝐼𝑋) ∈ 𝐵)
 
Theoremdrnginvrn0 19451 The multiplicative inverse in a division ring is nonzero. (recne0 11300 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝐼𝑋) ≠ 0 )
 
Theoremdrnginvrl 19452 Property of the multiplicative inverse in a division ring. (recid2 11302 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → ((𝐼𝑋) · 𝑋) = 1 )
 
Theoremdrnginvrr 19453 Property of the multiplicative inverse in a division ring. (recid 11301 analog.) (Contributed by NM, 19-Apr-2014.)
𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &    · = (.r𝑅)    &    1 = (1r𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝑋𝐵𝑋0 ) → (𝑋 · (𝐼𝑋)) = 1 )
 
Theoremdrngmul0or 19454 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 19455 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 19456 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 ))
 
Theoremopprdrng 19457 The opposite of a division ring is also a division ring. (Contributed by NM, 18-Oct-2014.)
𝑂 = (oppr𝑅)       (𝑅 ∈ DivRing ↔ 𝑂 ∈ DivRing)
 
Theoremisdrngd 19458* Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a left-inverse 𝐼(𝑥). See isdrngd 19458 for the characterization using right-inverses. (Contributed by NM, 2-Aug-2013.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑0 = (0g𝑅))    &   (𝜑1 = (1r𝑅))    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝑥 · 𝑦) ≠ 0 )    &   (𝜑10 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼0 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → (𝐼 · 𝑥) = 1 )       (𝜑𝑅 ∈ DivRing)
 
Theoremisdrngrd 19459* Properties that characterize a division ring among rings: it should be nonzero, have no nonzero zero-divisors, and every nonzero element 𝑥 should have a right-inverse 𝐼(𝑥). See isdrngd 19458 for the characterization using left-inverses. (Contributed by NM, 10-Aug-2013.)
(𝜑𝐵 = (Base‘𝑅))    &   (𝜑· = (.r𝑅))    &   (𝜑0 = (0g𝑅))    &   (𝜑1 = (1r𝑅))    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 ) ∧ (𝑦𝐵𝑦0 )) → (𝑥 · 𝑦) ≠ 0 )    &   (𝜑10 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼𝐵)    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → 𝐼0 )    &   ((𝜑 ∧ (𝑥𝐵𝑥0 )) → (𝑥 · 𝐼) = 1 )       (𝜑𝑅 ∈ DivRing)
 
Theoremdrngpropd 19460* If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing))
 
Theoremfldpropd 19461* If two structures have the same group components (properties), one is a field iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐾)𝑦) = (𝑥(.r𝐿)𝑦))       (𝜑 → (𝐾 ∈ Field ↔ 𝐿 ∈ Field))
 
10.4.2  Subrings of a ring
 
Syntaxcsubrg 19462 Extend class notation with all subrings of a ring.
class SubRing
 
Syntaxcrgspn 19463 Extend class notation with span of a set of elements over a ring.
class RingSpan
 
Definitiondf-subrg 19464* 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) ∣ ((𝑤s 𝑠) ∈ Ring ∧ (1r𝑤) ∈ 𝑠)})
 
Definitiondf-rgspn 19465* 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) ∣ 𝑠𝑡}))
 
Theoremissubrg 19466 The subring predicate. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Proof shortened by AV, 12-Oct-2020.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)       (𝐴 ∈ (SubRing‘𝑅) ↔ ((𝑅 ∈ Ring ∧ (𝑅s 𝐴) ∈ Ring) ∧ (𝐴𝐵1𝐴)))
 
Theoremsubrgss 19467 A subring is a subset. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝐵 = (Base‘𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 𝐴𝐵)
 
Theoremsubrgid 19468 Every ring is a subring of itself. (Contributed by Stefan O'Rear, 30-Nov-2014.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → 𝐵 ∈ (SubRing‘𝑅))
 
Theoremsubrgring 19469 A subring is a ring. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → 𝑆 ∈ Ring)
 
Theoremsubrgcrng 19470 A subring of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 10-Jan-2015.)
𝑆 = (𝑅s 𝐴)       ((𝑅 ∈ CRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → 𝑆 ∈ CRing)
 
Theoremsubrgrcl 19471 Reverse closure for a subring predicate. (Contributed by Mario Carneiro, 3-Dec-2014.)
(𝐴 ∈ (SubRing‘𝑅) → 𝑅 ∈ Ring)
 
Theoremsubrgsubg 19472 A subring is a subgroup. (Contributed by Mario Carneiro, 3-Dec-2014.)
(𝐴 ∈ (SubRing‘𝑅) → 𝐴 ∈ (SubGrp‘𝑅))
 
Theoremsubrg0 19473 A subring always has the same additive identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)    &    0 = (0g𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 0 = (0g𝑆))
 
Theoremsubrg1cl 19474 A subring contains the multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
1 = (1r𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 1𝐴)
 
Theoremsubrgbas 19475 Base set of a subring structure. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → 𝐴 = (Base‘𝑆))
 
Theoremsubrg1 19476 A subring always has the same multiplicative identity. (Contributed by Stefan O'Rear, 27-Nov-2014.)
𝑆 = (𝑅s 𝐴)    &    1 = (1r𝑅)       (𝐴 ∈ (SubRing‘𝑅) → 1 = (1r𝑆))
 
Theoremsubrgacl 19477 A subring is closed under addition. (Contributed by Mario Carneiro, 2-Dec-2014.)
+ = (+g𝑅)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ∈ 𝐴)
 
Theoremsubrgmcl 19478 A subgroup is closed under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
· = (.r𝑅)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝐴) → (𝑋 · 𝑌) ∈ 𝐴)
 
Theoremsubrgsubm 19479 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))
 
Theoremsubrgdvds 19480 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 = (𝑅s 𝐴)    &    = (∥r𝑅)    &   𝐸 = (∥r𝑆)       (𝐴 ∈ (SubRing‘𝑅) → 𝐸 )
 
Theoremsubrguss 19481 A unit of a subring is a unit of the parent ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (Unit‘𝑅)    &   𝑉 = (Unit‘𝑆)       (𝐴 ∈ (SubRing‘𝑅) → 𝑉𝑈)
 
Theoremsubrginv 19482 A subring always has the same inversion function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &   𝐼 = (invr𝑅)    &   𝑈 = (Unit‘𝑆)    &   𝐽 = (invr𝑆)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝑈) → (𝐼𝑋) = (𝐽𝑋))
 
Theoremsubrgdv 19483 A subring always has the same division function, for elements that are invertible. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &    / = (/r𝑅)    &   𝑈 = (Unit‘𝑆)    &   𝐸 = (/r𝑆)       ((𝐴 ∈ (SubRing‘𝑅) ∧ 𝑋𝐴𝑌𝑈) → (𝑋 / 𝑌) = (𝑋𝐸𝑌))
 
Theoremsubrgunit 19484 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 = (𝑅s 𝐴)    &   𝑈 = (Unit‘𝑅)    &   𝑉 = (Unit‘𝑆)    &   𝐼 = (invr𝑅)       (𝐴 ∈ (SubRing‘𝑅) → (𝑋𝑉 ↔ (𝑋𝑈𝑋𝐴 ∧ (𝐼𝑋) ∈ 𝐴)))
 
Theoremsubrgugrp 19485 The units of a subring form a subgroup of the unit group of the original ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &   𝑈 = (Unit‘𝑅)    &   𝑉 = (Unit‘𝑆)    &   𝐺 = ((mulGrp‘𝑅) ↾s 𝑈)       (𝐴 ∈ (SubRing‘𝑅) → 𝑉 ∈ (SubGrp‘𝐺))
 
Theoremissubrg2 19486* Characterize the subrings of a ring by closure properties. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝐵 = (Base‘𝑅)    &    1 = (1r𝑅)    &    · = (.r𝑅)       (𝑅 ∈ Ring → (𝐴 ∈ (SubRing‘𝑅) ↔ (𝐴 ∈ (SubGrp‘𝑅) ∧ 1𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥 · 𝑦) ∈ 𝐴)))
 
Theoremopprsubrg 19487 Being a subring is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
𝑂 = (oppr𝑅)       (SubRing‘𝑅) = (SubRing‘𝑂)
 
Theoremsubrgint 19488 The intersection of a nonempty collection of subrings is a subring. (Contributed by Stefan O'Rear, 30-Nov-2014.) (Revised by Mario Carneiro, 7-Dec-2014.)
((𝑆 ⊆ (SubRing‘𝑅) ∧ 𝑆 ≠ ∅) → 𝑆 ∈ (SubRing‘𝑅))
 
Theoremsubrgin 19489 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)) → (𝐴𝐵) ∈ (SubRing‘𝑅))
 
Theoremsubrgmre 19490 The subrings of a ring are a Moore system. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝐵 = (Base‘𝑅)       (𝑅 ∈ Ring → (SubRing‘𝑅) ∈ (Moore‘𝐵))
 
Theoremissubdrg 19491* Characterize the subfields of a division ring. (Contributed by Mario Carneiro, 3-Dec-2014.)
𝑆 = (𝑅s 𝐴)    &    0 = (0g𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅)) → (𝑆 ∈ DivRing ↔ ∀𝑥 ∈ (𝐴 ∖ { 0 })(𝐼𝑥) ∈ 𝐴))
 
Theoremsubsubrg 19492 A subring of a subring is a subring. (Contributed by Mario Carneiro, 4-Dec-2014.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → (𝐵 ∈ (SubRing‘𝑆) ↔ (𝐵 ∈ (SubRing‘𝑅) ∧ 𝐵𝐴)))
 
Theoremsubsubrg2 19493 The set of subrings of a subring are the smaller subrings. (Contributed by Stefan O'Rear, 9-Mar-2015.)
𝑆 = (𝑅s 𝐴)       (𝐴 ∈ (SubRing‘𝑅) → (SubRing‘𝑆) = ((SubRing‘𝑅) ∩ 𝒫 𝐴))
 
Theoremissubrg3 19494 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))))
 
Theoremresrhm 19495 Restriction of a ring homomorphism to a subring is a homomorphism. (Contributed by Mario Carneiro, 12-Mar-2015.)
𝑈 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 RingHom 𝑇) ∧ 𝑋 ∈ (SubRing‘𝑆)) → (𝐹𝑋) ∈ (𝑈 RingHom 𝑇))
 
Theoremrhmeql 19496 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 (𝐹𝐺) ∈ (SubRing‘𝑆))
 
Theoremrhmima 19497 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)) → (𝐹𝑋) ∈ (SubRing‘𝑁))
 
Theoremrnrhmsubrg 19498 The range of a ring homomorphism is a subring. (Contributed by SN, 18-Nov-2023.)
(𝐹 ∈ (𝑀 RingHom 𝑁) → ran 𝐹 ∈ (SubRing‘𝑁))
 
Theoremcntzsubr 19499 Centralizers in a ring are subrings. (Contributed by Stefan O'Rear, 6-Sep-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐵 = (Base‘𝑅)    &   𝑀 = (mulGrp‘𝑅)    &   𝑍 = (Cntz‘𝑀)       ((𝑅 ∈ Ring ∧ 𝑆𝐵) → (𝑍𝑆) ∈ (SubRing‘𝑅))
 
Theorempwsdiagrhm 19500* Diagonal homomorphism into a structure power (Rings). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐹 ∈ (𝑅 RingHom 𝑌))
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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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