P. 141 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 14001-14100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	ringlz 14001	The zero of a unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ X e. B ) -> ( .0. .x. X ) = .0. )
Theorem	ringrz 14002	The zero of a unital ring is a right-absorbing element. (Contributed by FL, 31-Aug-2009.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ X e. B ) -> ( X .x. .0. ) = .0. )
Theorem	ringlzd 14003	The zero of a unital ring is a left-absorbing element. (Contributed by SN, 7-Mar-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( .0. .x. X ) = .0. )
Theorem	ringrzd 14004	The zero of a unital ring is a right-absorbing element. (Contributed by SN, 7-Mar-2025.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .x. .0. ) = .0. )
Theorem	ringsrg 14005	Any ring is also a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)
|- ( R e. Ring -> R e. SRing )
Theorem	ring1eq0 14006	If one and zero are equal, then any two elements of a ring are equal. Alternately, every ring has one distinct from zero except the zero ring containing the single element { 0 }. (Contributed by Mario Carneiro, 10-Sep-2014.)
|- B = ( Base ` R )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ X e. B /\ Y e. B ) -> ( .1. = .0. -> X = Y ) )
Theorem	ringinvnz1ne0 14007*	In a unital ring, a left invertible element is different from zero iff .1. =/= .0.. (Contributed by FL, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> E. a e. B ( a .x. X ) = .1. )   =>    |- ( ph -> ( X =/= .0. <-> .1. =/= .0. ) )
Theorem	ringinvnzdiv 14008*	In a unital ring, a left invertible element is not a zero divisor. (Contributed by FL, 18-Apr-2010.) (Revised by Jeff Madsen, 18-Apr-2010.) (Revised by AV, 24-Aug-2021.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .1. = ( 1r ` R )   &    |- .0. = ( 0g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> E. a e. B ( a .x. X ) = .1. )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( X .x. Y ) = .0. <-> Y = .0. ) )
Theorem	ringnegl 14009	Negation in a ring is the same as left multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .1. = ( 1r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( ( N ` .1. ) .x. X ) = ( N ` X ) )
Theorem	ringnegr 14010	Negation in a ring is the same as right multiplication by -1. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .1. = ( 1r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .x. ( N ` .1. ) ) = ( N ` X ) )
Theorem	ringmneg1 14011	Negation of a product in a ring. (mulneg1 8537 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. Y ) = ( N ` ( X .x. Y ) ) )
Theorem	ringmneg2 14012	Negation of a product in a ring. (mulneg2 8538 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .x. ( N ` Y ) ) = ( N ` ( X .x. Y ) ) )
Theorem	ringm2neg 14013	Double negation of a product in a ring. (mul2neg 8540 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- N = ( invg ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( ( N ` X ) .x. ( N ` Y ) ) = ( X .x. Y ) )
Theorem	ringsubdi 14014	Ring multiplication distributes over subtraction. (subdi 8527 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( X .x. ( Y .- Z ) ) = ( ( X .x. Y ) .- ( X .x. Z ) ) )
Theorem	ringsubdir 14015	Ring multiplication distributes over subtraction. (subdir 8528 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .- = ( -g ` R )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- ( ph -> Z e. B )   =>    |- ( ph -> ( ( X .- Y ) .x. Z ) = ( ( X .x. Z ) .- ( Y .x. Z ) ) )
Theorem	mulgass2 14016	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 ) ) -> ( ( N .x. X ) .X. Y ) = ( N .x. ( X .X. Y ) ) )
Theorem	ring1 14017	The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)
|- M = { <. ( Base ` ndx ) , { Z } >. , <. ( +g ` ndx ) , { <. <. Z , Z >. , Z >. } >. , <. ( .r ` ndx ) , { <. <. Z , Z >. , Z >. } >. }   =>    |- ( Z e. V -> M e. Ring )
Theorem	ringn0 14018	The class of rings is not empty (it is also inhabited, as shown at ring1 14017). (Contributed by AV, 29-Apr-2019.)
|- Ring =/= (/)
Theorem	ringlghm 14019*	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.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( X .x. x ) ) e. ( R GrpHom R ) )
Theorem	ringrghm 14020*	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.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ X e. B ) -> ( x e. B |-> ( x .x. X ) ) e. ( R GrpHom R ) )
Theorem	ringressid 14021	A ring restricted to its base set is a ring. It will usually be the original ring exactly, of course, but to show that needs additional conditions such as those in strressid 13099. (Contributed by Jim Kingdon, 28-Feb-2025.)
|- B = ( Base ` G )   =>    |- ( G e. Ring -> ( G ↾s B ) e. Ring )
Theorem	imasring 14022*	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 ) ) )
Theorem	imasringf1 14023	The image of a ring under an injection is a ring. (Contributed by AV, 27-Feb-2025.)
|- U = ( F "s R )   &    |- V = ( Base ` R )   =>    |- ( ( F : V -1-1-> B /\ R e. Ring ) -> U e. Ring )
Theorem	qusring2 14024*	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 ) ) )
7.3.6  Opposite ring
Syntax	coppr 14025	The opposite ring operation.
class oppr
Definition	df-oppr 14026	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 ) >. ) )
Theorem	opprvalg 14027	Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- O = (oppr ` R )   =>    |- ( R e. V -> O = ( R sSet <. ( .r ` ndx ) , tpos .x. >. ) )
Theorem	opprmulfvalg 14028	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 )   =>    |- ( R e. V -> .xb = tpos .x. )
Theorem	opprmulg 14029	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 )   =>    |- ( ( R e. V /\ X e. W /\ Y e. U ) -> ( X .xb Y ) = ( Y .x. X ) )
Theorem	crngoppr 14030	In a commutative ring, the opposite ring is equivalent to the original ring. (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 ) )
Theorem	opprex 14031	Existence of the opposite ring. If you know that R is a ring, see opprring 14037. (Contributed by Jim Kingdon, 10-Jan-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> O e. _V )
Theorem	opprsllem 14032	Lemma for opprbasg 14033 and oppraddg 14034. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by AV, 6-Nov-2024.)
|- O = (oppr ` R )   &    |- ( E = Slot ( E ` ndx ) /\ ( E ` ndx ) e. NN )   &    |- ( E ` ndx ) =/= ( .r ` ndx )   =>    |- ( R e. V -> ( E ` R ) = ( E ` O ) )
Theorem	opprbasg 14033	Base set of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
|- O = (oppr ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. V -> B = ( Base ` O ) )
Theorem	oppraddg 14034	Addition operation of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) (Proof shortened by AV, 6-Nov-2024.)
|- O = (oppr ` R )   &    |- .+ = ( +g ` R )   =>    |- ( R e. V -> .+ = ( +g ` O ) )
Theorem	opprrng 14035	An opposite non-unital ring is a non-unital ring. (Contributed by AV, 15-Feb-2025.)
|- O = (oppr ` R )   =>    |- ( R e. Rng -> O e. Rng )
Theorem	opprrngbg 14036	A set is a non-unital ring if and only if its opposite is a non-unital ring. Bidirectional form of opprrng 14035. (Contributed by AV, 15-Feb-2025.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Rng <-> O e. Rng ) )
Theorem	opprring 14037	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 )
Theorem	opprringbg 14038	Bidirectional form of opprring 14037. (Contributed by Mario Carneiro, 6-Dec-2014.)
|- O = (oppr ` R )   =>    |- ( R e. V -> ( R e. Ring <-> O e. Ring ) )
Theorem	oppr0g 14039	Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- O = (oppr ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. V -> .0. = ( 0g ` O ) )
Theorem	oppr1g 14040	Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- O = (oppr ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. V -> .1. = ( 1r ` O ) )
Theorem	opprnegg 14041	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 = ( invg ` R )   =>    |- ( R e. V -> N = ( invg ` O ) )
Theorem	opprsubgg 14042	Being a subgroup is a symmetric property. (Contributed by Mario Carneiro, 6-Dec-2014.)
|- O = (oppr ` R )   =>    |- ( R e. V -> (SubGrp ` R ) = (SubGrp ` O ) )
Theorem	mulgass3 14043	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 ) ) )
7.3.7  Divisibility
Syntax	cdsr 14044	Ring divisibility relation.
class ||r
Syntax	cui 14045	Units in a ring.
class Unit
Syntax	cir 14046	Ring irreducibles.
class Irred
Definition	df-dvdsr 14047*	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 ) } )
Definition	df-unit 14048	Define the set of units in a ring, that is, all elements with a left and right multiplicative inverse. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- Unit = ( w e. _V |-> ( `' ( ( ||r ` w ) i^i ( ||r ` (oppr ` w ) ) ) " { ( 1r ` w ) } ) )
Definition	df-irred 14049*	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 } )
Theorem	reldvdsrsrg 14050	The divides relation is a relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Jim Kingdon, 24-Jan-2025.)
|- ( R e. SRing -> Rel ( ||r ` R ) )
Theorem	dvdsrvald 14051*	Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.) (Revised by Mario Carneiro, 6-Jan-2015.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .|| = ( ||r ` R ) )   &    |- ( ph -> R e. SRing )   &    |- ( ph -> .x. = ( .r ` R ) )   =>    |- ( ph -> .|| = { <. x , y >. | ( x e. B /\ E. z e. B ( z .x. x ) = y ) } )
Theorem	dvdsrd 14052*	Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .|| = ( ||r ` R ) )   &    |- ( ph -> R e. SRing )   &    |- ( ph -> .x. = ( .r ` R ) )   =>    |- ( ph -> ( X .|| Y <-> ( X e. B /\ E. z e. B ( z .x. X ) = Y ) ) )
Theorem	dvdsr2d 14053*	Value of the divides relation. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .|| = ( ||r ` R ) )   &    |- ( ph -> R e. SRing )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( X .|| Y <-> E. z e. B ( z .x. X ) = Y ) )
Theorem	dvdsrmuld 14054	A left-multiple of X is divisible by X. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .|| = ( ||r ` R ) )   &    |- ( ph -> R e. SRing )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> X .|| ( Y .x. X ) )
Theorem	dvdsrcld 14055	Closure of a dividing element. (Contributed by Mario Carneiro, 5-Dec-2014.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .|| = ( ||r ` R ) )   &    |- ( ph -> R e. SRing )   &    |- ( ph -> X .|| Y )   =>    |- ( ph -> X e. B )
Theorem	dvdsrex 14056	Existence of the divisibility relation. (Contributed by Jim Kingdon, 28-Jan-2025.)
|- ( R e. SRing -> ( ||r ` R ) e. _V )
Theorem	dvdsrcl2 14057	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 )
Theorem	dvdsrid 14058	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 )
Theorem	dvdsrtr 14059	Divisibility is transitive. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- B = ( Base ` R )   &    |- .|| = ( ||r ` R )   =>    |- ( ( R e. Ring /\ Y .|| Z /\ Z .|| X ) -> Y .|| X )
Theorem	dvdsrmul1 14060	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 ) )
Theorem	dvdsrneg 14061	An element divides its negative. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- B = ( Base ` R )   &    |- .|| = ( ||r ` R )   &    |- N = ( invg ` R )   =>    |- ( ( R e. Ring /\ X e. B ) -> X .|| ( N ` X ) )
Theorem	dvdsr01 14062	In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning.) (Contributed by Stefan O'Rear, 29-Mar-2015.)
|- B = ( Base ` R )   &    |- .|| = ( ||r ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ X e. B ) -> X .|| .0. )
Theorem	dvdsr02 14063	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. ) )
Theorem	isunitd 14064	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.)
|- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> .1. = ( 1r ` R ) )   &    |- ( ph -> .|| = ( ||r ` R ) )   &    |- ( ph -> S = (oppr ` R ) )   &    |- ( ph -> E = ( ||r ` S ) )   &    |- ( ph -> R e. SRing )   =>    |- ( ph -> ( X e. U <-> ( X .|| .1. /\ X E .1. ) ) )
Theorem	1unit 14065	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 )
Theorem	unitcld 14066	A unit is an element of the base set. (Contributed by Mario Carneiro, 1-Dec-2014.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> R e. SRing )   &    |- ( ph -> X e. U )   =>    |- ( ph -> X e. B )
Theorem	unitssd 14067	The set of units is contained in the base set. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> R e. SRing )   =>    |- ( ph -> U C_ B )
Theorem	opprunitd 14068	Being a unit is a symmetric property, so it transfers to the opposite ring. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> S = (oppr ` R ) )   &    |- ( ph -> R e. Ring )   =>    |- ( ph -> U = (Unit ` S ) )
Theorem	crngunit 14069	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. ) )
Theorem	dvdsunit 14070	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 )
Theorem	unitmulcl 14071	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 )
Theorem	unitmulclb 14072	Reversal of unitmulcl 14071 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 ) ) )
Theorem	unitgrpbasd 14073	The base set of the group of units. (Contributed by Mario Carneiro, 25-Dec-2014.)
|- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> G = ( (mulGrp ` R ) ↾s U ) )   &    |- ( ph -> R e. SRing )   =>    |- ( ph -> U = ( Base ` G ) )
Theorem	unitgrp 14074	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 )
Theorem	unitabl 14075	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 )
Theorem	unitgrpid 14076	The identity of the group of units of a ring is the ring unity. (Contributed by Mario Carneiro, 2-Dec-2014.)
|- U = (Unit ` R )   &    |- G = ( (mulGrp ` R ) ↾s U )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. Ring -> .1. = ( 0g ` G ) )
Theorem	unitsubm 14077	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 ) )
Syntax	cinvr 14078	Extend class notation with multiplicative inverse.
class invr
Definition	df-invr 14079	Define multiplicative inverse. (Contributed by NM, 21-Sep-2011.)
|- invr = ( r e. _V |-> ( invg ` ( (mulGrp ` r ) ↾s (Unit ` r ) ) ) )
Theorem	invrfvald 14080	Multiplicative inverse function for a ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.)
|- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> G = ( (mulGrp ` R ) ↾s U ) )   &    |- ( ph -> I = ( invr ` R ) )   &    |- ( ph -> R e. Ring )   =>    |- ( ph -> I = ( invg ` G ) )
Theorem	unitinvcl 14081	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 )
Theorem	unitinvinv 14082	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 )
Theorem	ringinvcl 14083	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 )
Theorem	unitlinv 14084	A unit times its inverse is the ring unity. (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. )
Theorem	unitrinv 14085	A unit times its inverse is the ring unity. (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. )
Theorem	1rinv 14086	The inverse of the ring unity is the ring unity. (Contributed by Mario Carneiro, 18-Jun-2015.)
|- I = ( invr ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( R e. Ring -> ( I ` .1. ) = .1. )
Theorem	0unit 14087	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. ) )
Theorem	unitnegcl 14088	The negative of a unit is a unit. (Contributed by Mario Carneiro, 4-Dec-2014.)
|- U = (Unit ` R )   &    |- N = ( invg ` R )   =>    |- ( ( R e. Ring /\ X e. U ) -> ( N ` X ) e. U )
Syntax	cdvr 14089	Extend class notation with ring division.
class /r
Definition	df-dvr 14090*	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 ) ) ) )
Theorem	dvrfvald 14091*	Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> I = ( invr ` R ) )   &    |- ( ph -> ./ = (/r ` R ) )   &    |- ( ph -> R e. SRing )   =>    |- ( ph -> ./ = ( x e. B , y e. U |-> ( x .x. ( I ` y ) ) ) )
Theorem	dvrvald 14092	Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
|- ( ph -> B = ( Base ` R ) )   &    |- ( ph -> .x. = ( .r ` R ) )   &    |- ( ph -> U = (Unit ` R ) )   &    |- ( ph -> I = ( invr ` R ) )   &    |- ( ph -> ./ = (/r ` R ) )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. U )   =>    |- ( ph -> ( X ./ Y ) = ( X .x. ( I ` Y ) ) )
Theorem	dvrcl 14093	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 )
Theorem	unitdvcl 14094	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 )
Theorem	dvrid 14095	A ring element divided by itself is the ring unity. (dividap 8844 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. )
Theorem	dvr1 14096	A ring element divided by the ring unity is itself. (div1 8846 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 )
Theorem	dvrass 14097	An associative law for division. (divassap 8833 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 ) ) )
Theorem	dvrcan1 14098	A cancellation law for division. (divcanap1 8824 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 )
Theorem	dvrcan3 14099	A cancellation law for division. (divcanap3 8841 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 )
Theorem	dvreq1 14100	Equality in terms of ratio equal to ring unity. (diveqap1 8848 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 ) )