HomeHome Metamath Proof Explorer
Theorem List (p. 155 of 322)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21498)
  Hilbert Space Explorer  Hilbert Space Explorer
(21499-23021)
  Users' Mathboxes  Users' Mathboxes
(23022-32154)
 

Theorem List for Metamath Proof Explorer - 15401-15500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempwsmgp 15401 The multiplicative group of the power structure resembles the power of the multiplicative group. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  M  =  (mulGrp `  R )   &    |-  Z  =  ( M  ^s  I )   &    |-  N  =  (mulGrp `  Y )   &    |-  B  =  (
 Base `  N )   &    |-  C  =  ( Base `  Z )   &    |-  .+  =  ( +g  `  N )   &    |-  .+b  =  ( +g  `  Z )   =>    |-  (
 ( R  e.  V  /\  I  e.  W )  ->  ( B  =  C  /\  .+  =  .+b  )
 )
 
Theoremimasrng 15402* The image structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .+  b )
 )  =  ( F `
  ( p  .+  q ) ) ) )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .x.  b )
 )  =  ( F `
  ( p  .x.  q ) ) ) )   &    |-  ( ph  ->  R  e.  Ring )   =>    |-  ( ph  ->  ( U  e.  Ring  /\  ( F `  .1.  )  =  ( 1r `  U ) ) )
 
Theoremdivsrng2 15403* The quotient structure of a ring is a ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |- 
 .+  =  ( +g  `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  ( ( a  .~  p  /\  b  .~  q ) 
 ->  ( a  .+  b
 )  .~  ( p  .+  q ) ) )   &    |-  ( ph  ->  ( (
 a  .~  p  /\  b  .~  q )  ->  ( a  .x.  b ) 
 .~  ( p  .x.  q ) ) )   &    |-  ( ph  ->  R  e.  Ring
 )   =>    |-  ( ph  ->  ( U  e.  Ring  /\  [  .1.  ]  .~  =  ( 1r `  U ) ) )
 
10.4.3  Opposite ring
 
Syntaxcoppr 15404 The opposite ring operation.
 class oppr
 
Definitiondf-oppr 15405 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  =  ( f  e.  _V  |->  ( f sSet  <. ( .r
 `  ndx ) , tpos  ( .r `  f ) >. ) )
 
Theoremopprval 15406 Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   =>    |-  O  =  ( R sSet  <. ( .r `  ndx ) , tpos  .x.  >. )
 
Theoremopprmulfval 15407 Value of the multiplication operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  .xb  = tpos  .x.
 
Theoremopprmul 15408 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.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  ( X  .xb  Y )  =  ( Y 
 .x.  X )
 
Theoremcrngoppr 15409 In a commutative ring, the opposite ring is equivalent to the original ring (for theorems like unitpropd 15479). (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  O  =  (oppr `  R )   &    |-  .xb  =  ( .r `  O )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( X 
 .xb  Y ) )
 
Theoremopprlem 15410 Lemma for opprbas 15411 and oppradd 15412. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  <  3   =>    |-  ( E `  R )  =  ( E `  O )
 
Theoremopprbas 15411 Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |-  B  =  ( Base `  R )   =>    |-  B  =  ( Base `  O )
 
Theoremoppradd 15412 Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .+  =  ( +g  `  R )   =>    |- 
 .+  =  ( +g  `  O )
 
Theoremopprrng 15413 An opposite ring is a ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Aug-2015.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  Ring  ->  O  e.  Ring )
 
Theoremopprrngb 15414 Bidirectional form of opprrng 15413. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  ( R  e.  Ring  <->  O  e.  Ring )
 
Theoremoppr0 15415 Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |- 
 .0.  =  ( 0g `  O )
 
Theoremoppr1 15416 Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  O  =  (oppr `  R )   &    |- 
 .1.  =  ( 1r `  R )   =>    |- 
 .1.  =  ( 1r `  O )
 
Theoremopprneg 15417 The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  O  =  (oppr `  R )   &    |-  N  =  ( inv
 g `  R )   =>    |-  N  =  ( inv g `  O )
 
Theoremopprsubg 15418 Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
 |-  O  =  (oppr `  R )   =>    |-  (SubGrp `  R )  =  (SubGrp `  O )
 
Theoremmulgass3 15419 An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  (.g `  R )   &    |-  .X.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( N  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( X  .X.  ( N 
 .x.  Y ) )  =  ( N  .x.  ( X  .X.  Y ) ) )
 
10.4.4  Divisibility
 
Syntaxcdsr 15420 Ring divides relation.
 class  ||r
 
Syntaxcui 15421 Ring unit.
 class Unit
 
Syntaxcir 15422 Ring irreducibles.
 class Irred
 
Definitiondf-dvdsr 15423* Define the (right) divisibility relation in a ring. Access to the left divisibility relation is available through  ( ||r `
 (oppr
`  R ) ). (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  ||r  =  ( w  e.  _V  |->  {
 <. x ,  y >.  |  ( x  e.  ( Base `  w )  /\  E. z  e.  ( Base `  w ) ( z ( .r `  w ) x )  =  y ) } )
 
Definitiondf-unit 15424 Define the set of units in a ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |- Unit  =  ( w  e.  _V  |->  ( `' ( ( ||r
 `  w )  i^i  ( ||r
 `  (oppr `  w ) ) )
 " { ( 1r
 `  w ) }
 ) )
 
Definitiondf-irred 15425* Define the set of irreducible elements in a ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |- Irred  =  ( w  e.  _V  |->  [_ ( ( Base `  w )  \  (Unit `  w ) )  /  b ]_ { z  e.  b  |  A. x  e.  b  A. y  e.  b  ( x ( .r `  w ) y )  =/=  z } )
 
Theoremreldvdsr 15426 The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  .||  =  ( ||r
 `  R )   =>    |-  Rel  .||
 
Theoremdvdsrval 15427* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  .||  =  { <. x ,  y >.  |  ( x  e.  B  /\  E. z  e.  B  (
 z  .x.  x )  =  y ) }
 
Theoremdvdsr 15428* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( X  .||  Y  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  Y ) )
 
Theoremdvdsr2 15429* Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( X  e.  B  ->  ( X  .||  Y  <->  E. z  e.  B  ( z  .x.  X )  =  Y ) )
 
Theoremdvdsrmul 15430 A left-multiple of  X is divisible by  X. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X  .||  ( Y 
 .x.  X ) )
 
Theoremdvdsrcl 15431 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( X  .||  Y  ->  X  e.  B )
 
Theoremdvdsrcl2 15432 Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  .||  Y )  ->  Y  e.  B )
 
Theoremdvdsrid 15433 An element in a (unital) ring divides itself. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  X  .||  X )
 
Theoremdvdsrtr 15434 Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  Ring  /\  Y  .||  Z  /\  Z  .||  X )  ->  Y  .||  X )
 
Theoremdvdsrmul1 15435 The divisibility relation is preserved under right-multiplication. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  Z  e.  B  /\  X  .||  Y )  ->  ( X  .x.  Z )  .||  ( Y  .x.  Z ) )
 
Theoremdvdsrneg 15436 An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |-  N  =  ( inv
 g `  R )   =>    |-  (
 ( R  e.  Ring  /\  X  e.  B ) 
 ->  X  .||  ( N `  X ) )
 
Theoremdvdsr01 15437 In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning, see df-rlreg 16024.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  X  .||  .0.  )
 
Theoremdvdsr02 15438 Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.)
 |-  B  =  ( Base `  R )   &    |-  .||  =  ( ||r `  R )   &    |- 
 .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  (  .0.  .||  X  <->  X  =  .0.  ) )
 
Theoremisunit 15439 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.  =  ( 1r `  R )   &    |-  .|| 
 =  ( ||r
 `  R )   &    |-  S  =  (oppr `  R )   &    |-  E  =  (
 ||r `  S )   =>    |-  ( X  e.  U  <->  ( X  .||  .1.  /\  X E  .1.  ) )
 
Theorem1unit 15440 The multiplicative identity is a unit. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  .1. 
 e.  U )
 
Theoremunitcl 15441 A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( X  e.  U  ->  X  e.  B )
 
Theoremunitss 15442 The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   =>    |-  U  C_  B
 
Theoremopprunit 15443 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  =  (oppr `  R )   =>    |-  U  =  (Unit `  S )
 
Theoremcrngunit 15444 Property of being a unit in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  U  =  (Unit `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .|| 
 =  ( ||r
 `  R )   =>    |-  ( R  e.  CRing  ->  ( X  e.  U  <->  X  .||  .1.  ) )
 
Theoremdvdsunit 15445 A divisor of a unit is a unit. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  U  =  (Unit `  R )   &    |-  .||  =  ( ||r `  R )   =>    |-  ( ( R  e.  CRing  /\  Y  .||  X  /\  X  e.  U )  ->  Y  e.  U )
 
Theoremunitmulcl 15446 The product of units is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  U )
 
Theoremunitmulclb 15447 Reversal of unitmulcl 15446 in a commutative ring. (Contributed by Mario Carneiro, 18-Apr-2016.)
 |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  B  =  ( Base `  R )   =>    |-  (
 ( R  e.  CRing  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .x.  Y )  e.  U  <->  ( X  e.  U  /\  Y  e.  U ) ) )
 
Theoremunitgrpbas 15448 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 15449 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  e.  Ring  ->  G  e.  Grp )
 
Theoremunitabl 15450 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  e.  CRing  ->  G  e.  Abel )
 
Theoremunitgrpid 15451 The identity of the multiplicative group is  1r. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  G  =  ( (mulGrp `  R )s  U )   &    |- 
 .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  .1.  =  ( 0g `  G ) )
 
Theoremunitsubm 15452 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  e.  Ring  ->  U  e.  (SubMnd `  M ) )
 
Syntaxcinvr 15453 Extend class notation with multiplicative inverse.
 class  invr
 
Definitiondf-invr 15454 Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
 |- 
 invr  =  ( r  e.  _V  |->  ( inv g `  ( (mulGrp `  r
 )s  (Unit `  r )
 ) ) )
 
Theoreminvrfval 15455 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  =  ( invr `  R )   =>    |-  I  =  ( inv
 g `  G )
 
Theoremunitinvcl 15456 The inverse of a unit exists and is a unit. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( I `
  X )  e.  U )
 
Theoremunitinvinv 15457 The inverse of the inverse of a unit is the same element. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U )  ->  ( I `
  ( I `  X ) )  =  X )
 
Theoremrnginvcl 15458 The inverse of a unit is an element of the ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  B  =  ( Base `  R )   =>    |-  (
 ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( I `  X )  e.  B )
 
Theoremunitlinv 15459 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( ( I `  X )  .x.  X )  =  .1.  )
 
Theoremunitrinv 15460 A unit times its inverse is the identity. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  I  =  (
 invr `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( X  .x.  ( I `  X ) )  =  .1.  )
 
Theorem1rinv 15461 The inverse of the identity is the identity. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  I  =  ( invr `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  ( I `  .1.  )  =  .1.  )
 
Theorem0unit 15462 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.  =  ( 0g `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( R  e.  Ring  ->  (  .0.  e.  U  <->  .1.  =  .0.  )
 )
 
Theoremunitnegcl 15463 The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  U  =  (Unit `  R )   &    |-  N  =  ( inv g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( N `  X )  e.  U )
 
Syntaxcdvr 15464 Extend class notation with ring division.
 class /r
 
Definitiondf-dvr 15465* Define ring division. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |- /r  =  ( r  e.  _V  |->  ( x  e.  ( Base `  r ) ,  y  e.  (Unit `  r )  |->  ( x ( .r `  r
 ) ( ( invr `  r ) `  y
 ) ) ) )
 
Theoremdvrfval 15466* Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  ( invr `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  ( I `  y ) ) )
 
Theoremdvrval 15467 Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  ( invr `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y )  =  ( X  .x.  ( I `  Y ) ) )
 
Theoremdvrcl 15468 Closure of division operation. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y )  e.  B )
 
Theoremunitdvcl 15469 The units are closed under division. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  U )  ->  ( X  ./  Y )  e.  U )
 
Theoremdvrid 15470 An cancellation law for division. (divid 9451 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( X  ./  X )  =  .1.  )
 
Theoremdvr1 15471 An cancellation law for division. (div1 9453 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  ./  .1.  )  =  X )
 
Theoremdvrass 15472 An associative law for division. (divass 9442 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  U ) )  ->  ( ( X  .x.  Y )  ./  Z )  =  ( X  .x.  ( Y  ./  Z ) ) )
 
Theoremdvrcan1 15473 A cancellation law for division. (divcan1 9433 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( X 
 ./  Y )  .x.  Y )  =  X )
 
Theoremdvrcan3 15474 A cancellation law for division. (divcan3 9448 analog.) (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( X 
 .x.  Y )  ./  Y )  =  X )
 
Theoremdvreq1 15475 A cancellation law for division. (diveq1 9454 analog.) (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  Y  e.  U )  ->  ( ( X  ./  Y )  =  .1.  <->  X  =  Y ) )
 
Theoremrnginvdv 15476 Write the inverse function in terms of division. (Contributed by Mario Carneiro, 2-Jul-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .1.  =  ( 1r `  R )   &    |-  I  =  ( invr `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( I `  X )  =  (  .1.  ./  X ) )
 
Theoremrngidpropd 15477* The ring identity depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( 1r `  K )  =  ( 1r `  L ) )
 
Theoremdvdsrpropd 15478* The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( ||r `  K )  =  (
 ||r `  L ) )
 
Theoremunitpropd 15479* The set of units depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  (Unit `  K )  =  (Unit `  L ) )
 
Theoreminvrpropd 15480* 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.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )   =>    |-  ( ph  ->  ( invr `  K )  =  ( invr `  L )
 )
 
Theoremisirred 15481* 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  =  ( B  \  U )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( X  e.  I  <->  ( X  e.  N  /\  A. x  e.  N  A. y  e.  N  ( x  .x.  y )  =/= 
 X ) )
 
Theoremisnirred 15482* 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  =  ( B  \  U )   &    |-  .x. 
 =  ( .r `  R )   =>    |-  ( X  e.  B  ->  ( -.  X  e.  I 
 <->  ( X  e.  U  \/  E. x  e.  N  E. y  e.  N  ( x  .x.  y )  =  X ) ) )
 
Theoremisirred2 15483* Expand out the set differences from isirred 15481. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  I  =  (Irred `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( X  e.  I 
 <->  ( X  e.  B  /\  -.  X  e.  U  /\  A. x  e.  B  A. y  e.  B  ( ( x  .x.  y
 )  =  X  ->  ( x  e.  U  \/  y  e.  U )
 ) ) )
 
Theoremopprirred 15484 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  =  (oppr `  R )   &    |-  I  =  (Irred `  R )   =>    |-  I  =  (Irred `  S )
 
Theoremirredn0 15485 The additive identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I ) 
 ->  X  =/=  .0.  )
 
Theoremirredcl 15486 An irreducible element is in the ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  B  =  (
 Base `  R )   =>    |-  ( X  e.  I  ->  X  e.  B )
 
Theoremirrednu 15487 An irreducible element is not a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  U  =  (Unit `  R )   =>    |-  ( X  e.  I  ->  -.  X  e.  U )
 
Theoremirredn1 15488 The multiplicative identity is not irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  .1.  =  ( 1r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I ) 
 ->  X  =/=  .1.  )
 
Theoremirredrmul 15489 The product of an irreducible element and a unit is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I  /\  Y  e.  U )  ->  ( X  .x.  Y )  e.  I )
 
Theoremirredlmul 15490 The product of a unit and an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  U  =  (Unit `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  U  /\  Y  e.  I
 )  ->  ( X  .x.  Y )  e.  I
 )
 
Theoremirredmul 15491 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 )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( X  e.  B  /\  Y  e.  B  /\  ( X  .x.  Y )  e.  I )  ->  ( X  e.  U  \/  Y  e.  U ) )
 
Theoremirredneg 15492 The negative of an irreducible element is irreducible. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  N  =  ( inv g `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  I ) 
 ->  ( N `  X )  e.  I )
 
Theoremirrednegb 15493 An element is irreducible iff its negative is. (Contributed by Mario Carneiro, 4-Dec-2014.)
 |-  I  =  (Irred `  R )   &    |-  N  =  ( inv g `  R )   &    |-  B  =  ( Base `  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B ) 
 ->  ( X  e.  I  <->  ( N `  X )  e.  I ) )
 
10.4.5  Ring homomorphisms
 
Syntaxcrh 15494 Ring homomorphisms.
 class RingHom
 
Syntaxcrs 15495 Ring isomorphisms.
 class RingIso
 
Definitiondf-rnghom 15496* Define the set of ring homomorphisms from  r to  s. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |- RingHom  =  ( r  e.  Ring ,  s  e.  Ring  |->  [_ ( Base `  r )  /  v ]_ [_ ( Base `  s )  /  w ]_
 { f  e.  ( w  ^m  v )  |  ( ( f `  ( 1r `  r ) )  =  ( 1r
 `  s )  /\  A. x  e.  v  A. y  e.  v  (
 ( f `  ( x ( +g  `  r
 ) y ) )  =  ( ( f `
  x ) (
 +g  `  s )
 ( f `  y
 ) )  /\  (
 f `  ( x ( .r `  r ) y ) )  =  ( ( f `  x ) ( .r
 `  s ) ( f `  y ) ) ) ) }
 )
 
Definitiondf-rngiso 15497* Define the set of ring isomorphisms from  r to  s. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |- RingIso  =  ( r  e.  _V ,  s  e.  _V  |->  { f  e.  ( r RingHom  s )  |  `' f  e.  ( s RingHom  r ) } )
 
Theoremdfrhm2 15498* 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  e.  Ring ,  s  e.  Ring  |->  ( ( r  GrpHom  s )  i^i  ( (mulGrp `  r
 ) MndHom  (mulGrp `  s )
 ) ) )
 
Theoremrhmrcl1 15499 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  R  e.  Ring )
 
Theoremrhmrcl2 15500 Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( R RingHom  S )  ->  S  e.  Ring )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 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 32101-32154
  Copyright terms: Public domain < Previous  Next >