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Theorem List for Metamath Proof Explorer - 15701-15800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrngrz 15701 The zero of a unital ring is a right absorbing element. (Contributed by FL, 31-Aug-2009.)

Theoremrng1eq0 15702 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 15703 Negation in a ring is the same as left multiplication by -1. (rngonegmn1l 26565 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngnegr 15704 Negation in a ring is the same as right multiplication by -1. (rngonegmn1r 26566 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngmneg1 15705 Negation of a product in a ring. (mulneg1 9470 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngmneg2 15706 Negation of a product in a ring. (mulneg2 9471 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngm2neg 15707 Double negation of a product in a ring. (mul2neg 9473 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)

Theoremrngsubdi 15708 Ring multiplication distributes over subtraction. (subdi 9467 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremrngsubdir 15709 Ring multiplication distributes over subtraction. (subdir 9468 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)

Theoremmulgass2 15710 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g

Theoremrnglghm 15711* 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 15712* 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 15713* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
g g

Theoremgsummulc2 15714* A finite ring sum multiplied by a constant. (Contributed by Mario Carneiro, 19-Dec-2014.)
g g

Theoremgsumdixp 15715* 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 15716 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 15717 A structure product of rings has closed binary operation. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprdsrngd 15718 A product of rings is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theoremprdscrngd 15719 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 15720 Value of the ring unit in a structure family product. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwsrng 15721 A structure power of a ring is a ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempws1 15722 Value of the ring unit in a structure power. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwscrng 15723 A structure power of a commutative ring is a commutative ring. (Contributed by Mario Carneiro, 11-Mar-2015.)
s

Theorempwsmgp 15724 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 15725* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s

Theoremdivsrng2 15726* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
s

10.4.3  Opposite ring

Syntaxcoppr 15727 The opposite ring operation.
oppr

Definitiondf-oppr 15728 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 15729 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr       sSet tpos

Theoremopprmulfval 15730 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr              tpos

Theoremopprmul 15731 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 15732 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 15802). (Contributed by Mario Carneiro, 14-Jun-2015.)
oppr

Theoremopprlem 15733 Lemma for opprbas 15734 and oppradd 15735. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr       Slot

Theoremopprbas 15734 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremoppradd 15735 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremopprrng 15736 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
oppr

Theoremopprrngb 15737 Bidirectional form of opprrng 15736. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr

Theoremoppr0 15738 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremoppr1 15739 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
oppr

Theoremopprneg 15740 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
oppr

Theoremopprsubg 15741 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
oppr       SubGrp SubGrp

Theoremmulgass3 15742 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
.g

10.4.4  Divisibility

Syntaxcdsr 15743 Ring divides relation.
r

Syntaxcui 15744 Ring unit.
Unit

Syntaxcir 15745 Ring irreducibles.
Irred

Definitiondf-dvdsr 15746* 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 15747 Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit r roppr

Definitiondf-irred 15748* Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
Irred Unit

Theoremreldvdsr 15749 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrval 15750* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
r

Theoremdvdsr 15751* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsr2 15752* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrmul 15753 A left-multiple of is divisible by . (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrcl 15754 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
r

Theoremdvdsrcl2 15755 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
r

Theoremdvdsrid 15756 An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
r

Theoremdvdsrtr 15757 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrmul1 15758 The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsrneg 15759 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
r

Theoremdvdsr01 15760 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 16343.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
r

Theoremdvdsr02 15761 Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
r

Theoremisunit 15762 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 15763 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit

Theoremunitcl 15764 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
Unit

Theoremunitss 15765 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
Unit

Theoremopprunit 15766 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 15767 Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
Unit              r

Theoremdvdsunit 15768 A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
Unit       r

Theoremunitmulcl 15769 The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitmulclb 15770 Reversal of unitmulcl 15769 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
Unit

Theoremunitgrpbas 15771 The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
Unit       mulGrps

Theoremunitgrp 15772 The group of units is a group under multiplication. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit       mulGrps

Theoremunitabl 15773 The group of units of a commutative ring is abelian. (Contributed by Mario Carneiro, 19-Apr-2016.)
Unit       mulGrps

Theoremunitgrpid 15774 The identity of the multiplicative group is . (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit       mulGrps

Theoremunitsubm 15775 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 15776 Extend class notation with multiplicative inverse.

Definitiondf-invr 15777 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
mulGrps Unit

Theoreminvrfval 15778 Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
Unit       mulGrps

Theoremunitinvcl 15779 The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitinvinv 15780 The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit

Theoremrnginvcl 15781 The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitlinv 15782 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theoremunitrinv 15783 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
Unit

Theorem1rinv 15784 The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theorem0unit 15785 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 15786 The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit

Syntaxcdvr 15787 Extend class notation with ring division.
/r

Definitiondf-dvr 15788* Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
/r Unit

Theoremdvrfval 15789* Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Unit              /r

Theoremdvrval 15790 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Unit              /r

Theoremdvrcl 15791 Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.)
Unit       /r

Theoremunitdvcl 15792 The units are closed under division. (Contributed by Mario Carneiro, 2-Jul-2014.)
Unit       /r

Theoremdvrid 15793 A cancellation law for division. (divid 9705 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
Unit       /r

Theoremdvr1 15794 A cancellation law for division. (div1 9707 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
/r

Theoremdvrass 15795 An associative law for division. (divass 9696 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
Unit       /r

Theoremdvrcan1 15796 A cancellation law for division. (divcan1 9687 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Unit       /r

Theoremdvrcan3 15797 A cancellation law for division. (divcan3 9702 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.)
Unit       /r

Theoremdvreq1 15798 A cancellation law for division. (diveq1 9708 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.)
Unit       /r

Theoremrnginvdv 15799 Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
Unit       /r

Theoremrngidpropd 15800* The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)

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