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Theorem List for Metamath Proof Explorer - 15201-15300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremisrngid 15201* Properties showing that an element is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)

Theoremrngidss 15202 A subset of the multiplicative group has the multiplicative identity as its identity if the identity is in the subset. (Contributed by Mario Carneiro, 27-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
mulGrps

Theoremrngacl 15203 Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.)

Theoremrngcom 15204 Commutativity of the additive group of a ring. (See also lmodcom 15506.) (Contributed by Gérard Lang, 4-Dec-2014.)

Theoremrngabl 15205 A ring is an Abelian group. (Contributed by NM, 26-Aug-2011.)

Theoremrngcmn 15206 A ring is a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
CMnd

Theoremrngpropd 15207* If two structures have the same group components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 6-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)

Theoremcrngpropd 15208* If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.)

Theoremrngprop 15209 If two structures have the same ring components (properties), one is a ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.)

Theoremisrngd 15210* Properties that determine a ring. (Contributed by NM, 2-Aug-2013.)

Theoremiscrngd 15211* Properties that determine a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.)

Theoremrnglz 15212 The zero of a unital ring is a left absorbing element. (Contributed by FL, 31-Aug-2009.)

Theoremrngrz 15213 The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.)

Theoremrng1eq0 15214 If one and zero are equal, then any two elements of a ring are equal. Alternatively, every ring has one distinct from zero except the zero ring containing the single element . (Contributed by Mario Carneiro, 10-Sep-2014.)

Theoremrngnegl 15215 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 25746 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngnegr 15216 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 25747 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngmneg1 15217 Negation of a product in a ring. (mulneg1 9096 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngmneg2 15218 Negation of a product in a ring. (mulneg2 9097 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngm2neg 15219 Double negation of a product in a ring. (mul2neg 9099 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)

Theoremrngsubdi 15220 Ring multiplication distributes over subtraction. (subdi 9093 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngsubdir 15221 Ring multiplication distributes over subtraction. (subdir 9094 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremmulgass2 15222 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g

Theoremrnglghm 15223* Left-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)

Theoremrngrghm 15224* Right-multiplication in a ring by a fixed element of the ring is a group homomorphism. (It is not usually a ring homomorphism.) (Contributed by Mario Carneiro, 4-May-2015.)

Theoremgsummulc1 15225* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
g g

Theoremgsummulc2 15226* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
g g

Theoremgsumdixp 15227* Distribute a binary product of sums to a sum of binary products in a ring. (Contributed by Mario Carneiro, 8-Mar-2015.)
g g g

Theoremprdsmgp 15228 The multiplicative monoid of a product is the product of the multiplicative monoids of the factors. (Contributed by Mario Carneiro, 11-Mar-2015.)
s       mulGrp       smulGrp

Theoremprdsmulrcl 15229 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprdsrngd 15230 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprdscrngd 15231 A product of commutative rings is a commutative ring. Since the resulting ring will have zero divisors in all nontrivial cases, this cannot be strengthened much further. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprds1 15232 Value of the ring unit in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwsrng 15233 A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempws1 15234 Value of the ring unit in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwscrng 15235 A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwsmgp 15236 The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
s        mulGrp       s        mulGrp

Theoremimasrng 15237* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s

Theoremdivsrng2 15238* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s

10.4.3  Opposite ring

Syntaxcoppr 15239 The opposite ring operation.
oppr

Definitiondf-oppr 15240 Define an opposite ring, which is the same as the original ring but with multiplication written the other way around. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr sSet tpos

Theoremopprval 15241 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr       sSet tpos

Theoremopprmulfval 15242 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr              tpos

Theoremopprmul 15243 Value of the multiplication operation of an opposite ring. Hypotheses eliminated by a suggestion of Stefan O'Rear, 30-Aug-2015. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
oppr

Theoremcrngoppr 15244 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 15314). (Contributed by Mario Carneiro, 14-Jun-2015.)
oppr

Theoremopprlem 15245 Lemma for opprbas 15246 and oppradd 15247. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr       Slot

Theoremopprbas 15246 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremoppradd 15247 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremopprrng 15248 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
oppr

Theoremopprrngb 15249 Bidirectional form of opprrng 15248. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr

Theoremoppr0 15250 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremoppr1 15251 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremopprneg 15252 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
oppr

Theoremopprsubg 15253 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubGrp SubGrp

Theoremmulgass3 15254 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g

10.4.4  Divisibility

Syntaxcdsr 15255 Ring divides relation.
r

Syntaxcui 15256 Ring unit.
Unit

Syntaxcir 15257 Ring irreducibles.
Irred

Definitiondf-dvdsr 15258* Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through roppr. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Definitiondf-unit 15259 Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit r roppr

Definitiondf-irred 15260* Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred Unit

Theoremreldvdsr 15261 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrval 15262* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
r

Theoremdvdsr 15263* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsr2 15264* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrmul 15265 A left-multiple of is divisible by . (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrcl 15266 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
r

Theoremdvdsrcl2 15267 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
r

Theoremdvdsrid 15268 An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
r

Theoremdvdsrtr 15269 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrmul1 15270 The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrneg 15271 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsr01 15272 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 15856.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
r

Theoremdvdsr02 15273 Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
r

Theoremisunit 15274 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.)
Unit              r       oppr       r

Theorem1unit 15275 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit

Theoremunitcl 15276 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit

Theoremunitss 15277 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit

Theoremopprunit 15278 Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       oppr       Unit

Theoremcrngunit 15279 Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
Unit              r

Theoremdvdsunit 15280 A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
Unit       r

Theoremunitmulcl 15281 The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitmulclb 15282 Reversal of unitmulcl 15281 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
Unit

Theoremunitgrpbas 15283 The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
Unit       mulGrps

Theoremunitgrp 15284 The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit       mulGrps

Theoremunitabl 15285 The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.)
Unit       mulGrps

Theoremunitgrpid 15286 The identity of the multiplicative group is . (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit       mulGrps

Theoremunitsubm 15287 The group of units is a submonoid of the multiplicative monoid of the ring. (Contributed by Mario Carneiro, 18-Jun-2015.)
Unit       mulGrp       SubMnd

Syntaxcinvr 15288 Extend class notation with multiplicative inverse.

Definitiondf-invr 15289 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
mulGrps Unit

Theoreminvrfval 15290 Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
Unit       mulGrps

Theoremunitinvcl 15291 The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitinvinv 15292 The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit

Theoremrnginvcl 15293 The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitlinv 15294 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitrinv 15295 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theorem1rinv 15296 The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theorem0unit 15297 The additive identity is a unit if and only if , i.e. we are in the zero ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit

Theoremunitnegcl 15298 The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit

Syntaxcdvr 15299 Extend class notation with ring division.
/r

Definitiondf-dvr 15300* Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
/r Unit

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