P. 138 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 13701-13800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	rlmbasg 13701	Base set of the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( R e. V -> ( Base ` R ) = ( Base ` (ringLMod ` R ) ) )
Theorem	rlmplusgg 13702	Vector addition in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( R e. V -> ( +g ` R ) = ( +g ` (ringLMod ` R ) ) )
Theorem	rlm0g 13703	Zero vector in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- ( R e. V -> ( 0g ` R ) = ( 0g ` (ringLMod ` R ) ) )
Theorem	rlmsubg 13704	Subtraction in the ring module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
|- ( R e. V -> ( -g ` R ) = ( -g ` (ringLMod ` R ) ) )
Theorem	rlmmulrg 13705	Ring multiplication in the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- ( R e. V -> ( .r ` R ) = ( .r ` (ringLMod ` R ) ) )
Theorem	rlmscabas 13706	Scalars in the ring module have the same base set. (Contributed by Jim Kingdon, 29-Apr-2025.)
|- ( R e. X -> ( Base ` R ) = ( Base ` (Scalar ` (ringLMod ` R ) ) ) )
Theorem	rlmvscag 13707	Scalar multiplication in the ring module. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( R e. V -> ( .r ` R ) = ( .s ` (ringLMod ` R ) ) )
Theorem	rlmtopng 13708	Topology component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- ( R e. V -> ( TopOpen ` R ) = ( TopOpen ` (ringLMod ` R ) ) )
Theorem	rlmdsg 13709	Metric component of the ring module. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- ( R e. V -> ( dist ` R ) = ( dist ` (ringLMod ` R ) ) )
Theorem	rlmlmod 13710	The ring module is a module. (Contributed by Stefan O'Rear, 6-Dec-2014.)
|- ( R e. Ring -> (ringLMod ` R ) e. LMod )
Theorem	rlmvnegg 13711	Vector negation in the ring module. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 5-Jun-2015.)
|- ( R e. V -> ( invg ` R ) = ( invg ` (ringLMod ` R ) ) )
Theorem	ixpsnbasval 13712*	The value of an infinite Cartesian product of the base of a left module over a ring with a singleton. (Contributed by AV, 3-Dec-2018.)
|- ( ( R e. V /\ X e. W ) -> X_ x e. { X } ( Base ` ( ( { X } X. { (ringLMod ` R ) } ) ` x ) ) = { f | ( f Fn { X } /\ ( f ` X ) e. ( Base ` R ) ) } )
7.6.2  Ideals and spans
Syntax	clidl 13713	Ring left-ideal function.
class LIdeal
Syntax	crsp 13714	Ring span function.
class RSpan
Definition	df-lidl 13715	Define the class of left ideals of a given ring. An ideal is a submodule of the ring viewed as a module over itself. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- LIdeal = ( LSubSp o. ringLMod )
Definition	df-rsp 13716	Define the linear span function in a ring (Ideal generator). (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- RSpan = ( LSpan o. ringLMod )
Theorem	lidlvalg 13717	Value of the set of ring ideals. (Contributed by Stefan O'Rear, 31-Mar-2015.)
|- ( W e. V -> (LIdeal ` W ) = ( LSubSp ` (ringLMod ` W ) ) )
Theorem	rspvalg 13718	Value of the ring span function. (Contributed by Stefan O'Rear, 4-Apr-2015.)
|- ( W e. V -> (RSpan ` W ) = ( LSpan ` (ringLMod ` W ) ) )
Theorem	lidlex 13719	Existence of the set of left ideals. (Contributed by Jim Kingdon, 27-Apr-2025.)
|- ( W e. V -> (LIdeal ` W ) e. _V )
Theorem	rspex 13720	Existence of the ring span. (Contributed by Jim Kingdon, 25-Apr-2025.)
|- ( W e. V -> (RSpan ` W ) e. _V )
Theorem	lidlmex 13721	Existence of the set a left ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- I = (LIdeal ` W )   =>    |- ( U e. I -> W e. _V )
Theorem	lidlss 13722	An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015.)
|- B = ( Base ` W )   &    |- I = (LIdeal ` W )   =>    |- ( U e. I -> U C_ B )
Theorem	lidlssbas 13723	The base set of the restriction of the ring to a (left) ideal is a subset of the base set of the ring. (Contributed by AV, 17-Feb-2020.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   =>    |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
Theorem	lidlbas 13724	A (left) ideal of a ring is the base set of the restriction of the ring to this ideal. (Contributed by AV, 17-Feb-2020.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   =>    |- ( U e. L -> ( Base ` I ) = U )
Theorem	islidlm 13725*	Predicate of being a (left) ideal. (Contributed by Stefan O'Rear, 1-Apr-2015.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   =>    |- ( I e. U <-> ( I C_ B /\ E. j j e. I /\ A. x e. B A. a e. I A. b e. I ( ( x .x. a ) .+ b ) e. I ) )
Theorem	rnglidlmcl 13726	A (left) ideal containing the zero element is closed under left-multiplication by elements of the full non-unital ring. If the ring is not a unital ring, and the ideal does not contain the zero element of the ring, then the closure cannot be proven. (Contributed by AV, 18-Feb-2025.)
|- .0. = ( 0g ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- U = (LIdeal ` R )   =>    |- ( ( ( R e. Rng /\ I e. U /\ .0. e. I ) /\ ( X e. B /\ Y e. I ) ) -> ( X .x. Y ) e. I )
Theorem	dflidl2rng 13727*	Alternate (the usual textbook) definition of a (left) ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( x .x. y ) e. I ) )
Theorem	isridlrng 13728*	A right ideal is a left ideal of the opposite non-unital ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( y .x. x ) e. I ) )
Theorem	lidl0cl 13729	An ideal contains 0. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Ring /\ I e. U ) -> .0. e. I )
Theorem	lidlacl 13730	An ideal is closed under addition. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- .+ = ( +g ` R )   =>    |- ( ( ( R e. Ring /\ I e. U ) /\ ( X e. I /\ Y e. I ) ) -> ( X .+ Y ) e. I )
Theorem	lidlnegcl 13731	An ideal contains negatives. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- N = ( invg ` R )   =>    |- ( ( R e. Ring /\ I e. U /\ X e. I ) -> ( N ` X ) e. I )
Theorem	lidlsubg 13732	An ideal is a subgroup of the additive group. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- U = (LIdeal ` R )   =>    |- ( ( R e. Ring /\ I e. U ) -> I e. (SubGrp ` R ) )
Theorem	lidlsubcl 13733	An ideal is closed under subtraction. (Contributed by Stefan O'Rear, 28-Mar-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
|- U = (LIdeal ` R )   &    |- .- = ( -g ` R )   =>    |- ( ( ( R e. Ring /\ I e. U ) /\ ( X e. I /\ Y e. I ) ) -> ( X .- Y ) e. I )
Theorem	dflidl2 13734*	Alternate (the usual textbook) definition of a (left) ideal of a ring to be a subgroup of the additive group of the ring which is closed under left-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( x .x. y ) e. I ) ) )
Theorem	lidl0 13735	Every ring contains a zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> { .0. } e. U )
Theorem	lidl1 13736	Every ring contains a unit ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- U = (LIdeal ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. Ring -> B e. U )
Theorem	rspcl 13737	The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- K = (RSpan ` R )   &    |- B = ( Base ` R )   &    |- U = (LIdeal ` R )   =>    |- ( ( R e. Ring /\ G C_ B ) -> ( K ` G ) e. U )
Theorem	rspssid 13738	The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- K = (RSpan ` R )   &    |- B = ( Base ` R )   =>    |- ( ( R e. Ring /\ G C_ B ) -> G C_ ( K ` G ) )
Theorem	rsp0 13739	The span of the zero element is the zero ideal. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- K = (RSpan ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> ( K ` { .0. } ) = { .0. } )
Theorem	rspssp 13740	The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015.)
|- K = (RSpan ` R )   &    |- U = (LIdeal ` R )   =>    |- ( ( R e. Ring /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )
Theorem	lidlrsppropdg 13741*	The left ideals and ring span of a ring depend only on the ring components. Here W is expected to be either B (when closure is available) or _V (when strong equality is available). (Contributed by Mario Carneiro, 14-Jun-2015.)
|- ( ph -> B = ( Base ` K ) )   &    |- ( ph -> B = ( Base ` L ) )   &    |- ( ph -> B C_ W )   &    |- ( ( ph /\ ( x e. W /\ y e. W ) ) -> ( x ( +g ` K ) y ) = ( x ( +g ` L ) y ) )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) e. W )   &    |- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x ( .r ` K ) y ) = ( x ( .r ` L ) y ) )   &    |- ( ph -> K e. X )   &    |- ( ph -> L e. Y )   =>    |- ( ph -> ( (LIdeal ` K ) = (LIdeal ` L ) /\ (RSpan ` K ) = (RSpan ` L ) ) )
Theorem	rnglidlmmgm 13742	The multiplicative group of a (left) ideal of a non-unital ring is a magma. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption .0. e. U is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Mgm )
Theorem	rnglidlmsgrp 13743	The multiplicative group of a (left) ideal of a non-unital ring is a semigroup. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption .0. e. U is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   &    |- .0. = ( 0g ` R )   =>    |- ( ( R e. Rng /\ U e. L /\ .0. e. U ) -> (mulGrp ` I ) e. Smgrp )
Theorem	rnglidlrng 13744	A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption U e. (SubGrp ` R ) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)
|- L = (LIdeal ` R )   &    |- I = ( R ↾s U )   =>    |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )
7.6.3  Two-sided ideals and quotient rings
Syntax	c2idl 13745	Ring two-sided ideal function.
class 2Ideal
Definition	df-2idl 13746	Define the class of two-sided ideals of a ring. A two-sided ideal is a left ideal which is also a right ideal (or a left ideal over the opposite ring). (Contributed by Mario Carneiro, 14-Jun-2015.)
|- 2Ideal = ( r e. _V |-> ( (LIdeal ` r ) i^i (LIdeal ` (oppr ` r ) ) ) )
Theorem	2idlvalg 13747	Definition of a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- I = (LIdeal ` R )   &    |- O = (oppr ` R )   &    |- J = (LIdeal ` O )   &    |- T = (2Ideal ` R )   =>    |- ( R e. V -> T = ( I i^i J ) )
Theorem	2idlmex 13748	Existence of the set a two-sided ideal is built from (when the ideal is inhabited). (Contributed by Jim Kingdon, 18-Apr-2025.)
|- T = (2Ideal ` W )   =>    |- ( U e. T -> W e. _V )
Theorem	isridl 13749*	A right ideal is a left ideal of the opposite ring. This theorem shows that this definition corresponds to the usual textbook definition of a right ideal of a ring to be a subgroup of the additive group of the ring which is closed under right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( y .x. x ) e. I ) ) )
Theorem	2idlelb 13750	Membership in a two-sided ideal. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.)
|- I = (LIdeal ` R )   &    |- O = (oppr ` R )   &    |- J = (LIdeal ` O )   &    |- T = (2Ideal ` R )   =>    |- ( U e. T <-> ( U e. I /\ U e. J ) )
Theorem	2idllidld 13751	A two-sided ideal is a left ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   =>    |- ( ph -> I e. (LIdeal ` R ) )
Theorem	2idlridld 13752	A two-sided ideal is a right ideal. (Contributed by Thierry Arnoux, 9-Mar-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   &    |- O = (oppr ` R )   =>    |- ( ph -> I e. (LIdeal ` O ) )
Theorem	df2idl2rng 13753*	Alternate (the usual textbook) definition of a two-sided ideal of a non-unital ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 21-Mar-2025.)
|- U = (2Ideal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ I e. (SubGrp ` R ) ) -> ( I e. U <-> A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) )
Theorem	df2idl2 13754*	Alternate (the usual textbook) definition of a two-sided ideal of a ring to be a subgroup of the additive group of the ring which is closed under left- and right-multiplication by elements of the full ring. (Contributed by AV, 13-Feb-2025.) (Proof shortened by AV, 18-Apr-2025.)
|- U = (2Ideal ` R )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   =>    |- ( R e. Ring -> ( I e. U <-> ( I e. (SubGrp ` R ) /\ A. x e. B A. y e. I ( ( x .x. y ) e. I /\ ( y .x. x ) e. I ) ) ) )
Theorem	ridl0 13755	Every ring contains a zero right ideal. (Contributed by AV, 13-Feb-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> { .0. } e. U )
Theorem	ridl1 13756	Every ring contains a unit right ideal. (Contributed by AV, 13-Feb-2025.)
|- U = (LIdeal ` (oppr ` R ) )   &    |- B = ( Base ` R )   =>    |- ( R e. Ring -> B e. U )
Theorem	2idl0 13757	Every ring contains a zero two-sided ideal. (Contributed by AV, 13-Feb-2025.)
|- I = (2Ideal ` R )   &    |- .0. = ( 0g ` R )   =>    |- ( R e. Ring -> { .0. } e. I )
Theorem	2idl1 13758	Every ring contains a unit two-sided ideal. (Contributed by AV, 13-Feb-2025.)
|- I = (2Ideal ` R )   &    |- B = ( Base ` R )   =>    |- ( R e. Ring -> B e. I )
Theorem	2idlss 13759	A two-sided ideal is a subset of the base set. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 20-Feb-2025.) (Proof shortened by AV, 13-Mar-2025.)
|- B = ( Base ` W )   &    |- I = (2Ideal ` W )   =>    |- ( U e. I -> U C_ B )
Theorem	2idlbas 13760	The base set of a two-sided ideal as structure. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   &    |- J = ( R ↾s I )   &    |- B = ( Base ` J )   =>    |- ( ph -> B = I )
Theorem	2idlelbas 13761	The base set of a two-sided ideal as structure is a left and right ideal. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> I e. (2Ideal ` R ) )   &    |- J = ( R ↾s I )   &    |- B = ( Base ` J )   =>    |- ( ph -> ( B e. (LIdeal ` R ) /\ B e. (LIdeal ` (oppr ` R ) ) ) )
Theorem	rng2idlsubrng 13762	A two-sided ideal of a non-unital ring which is a non-unital ring is a subring of the ring. (Contributed by AV, 20-Feb-2025.) (Revised by AV, 11-Mar-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> ( R ↾s I ) e. Rng )   =>    |- ( ph -> I e. (SubRng ` R ) )
Theorem	rng2idlnsg 13763	A two-sided ideal of a non-unital ring which is a non-unital ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> ( R ↾s I ) e. Rng )   =>    |- ( ph -> I e. (NrmSGrp ` R ) )
Theorem	rng2idl0 13764	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a non-unital ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> ( R ↾s I ) e. Rng )   =>    |- ( ph -> ( 0g ` R ) e. I )
Theorem	rng2idlsubgsubrng 13765	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a subring of the ring. (Contributed by AV, 11-Mar-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> I e. (SubGrp ` R ) )   =>    |- ( ph -> I e. (SubRng ` R ) )
Theorem	rng2idlsubgnsg 13766	A two-sided ideal of a non-unital ring which is a subgroup of the ring is a normal subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> I e. (SubGrp ` R ) )   =>    |- ( ph -> I e. (NrmSGrp ` R ) )
Theorem	rng2idlsubg0 13767	The zero (additive identity) of a non-unital ring is an element of each two-sided ideal of the ring which is a subgroup of the ring. (Contributed by AV, 20-Feb-2025.)
|- ( ph -> R e. Rng )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> I e. (SubGrp ` R ) )   =>    |- ( ph -> ( 0g ` R ) e. I )
Theorem	2idlcpblrng 13768	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) Generalization for non-unital rings and two-sided ideals which are subgroups of the additive group of the non-unital ring. (Revised by AV, 23-Feb-2025.)
|- X = ( Base ` R )   &    |- E = ( R ~QG S )   &    |- I = (2Ideal ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) )
Theorem	2idlcpbl 13769	The coset equivalence relation for a two-sided ideal is compatible with ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) (Proof shortened by AV, 31-Mar-2025.)
|- X = ( Base ` R )   &    |- E = ( R ~QG S )   &    |- I = (2Ideal ` R )   &    |- .x. = ( .r ` R )   =>    |- ( ( R e. Ring /\ S e. I ) -> ( ( A E C /\ B E D ) -> ( A .x. B ) E ( C .x. D ) ) )
Theorem	qus2idrng 13770	The quotient of a non-unital ring modulo a two-sided ideal, which is a subgroup of the additive group of the non-unital ring, is a non-unital ring (qusring 13772 analog). (Contributed by AV, 23-Feb-2025.)
|- U = ( R /.s ( R ~QG S ) )   &    |- I = (2Ideal ` R )   =>    |- ( ( R e. Rng /\ S e. I /\ S e. (SubGrp ` R ) ) -> U e. Rng )
Theorem	qus1 13771	The multiplicative identity of the quotient ring. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- U = ( R /.s ( R ~QG S ) )   &    |- I = (2Ideal ` R )   &    |- .1. = ( 1r ` R )   =>    |- ( ( R e. Ring /\ S e. I ) -> ( U e. Ring /\ [ .1. ] ( R ~QG S ) = ( 1r ` U ) ) )
Theorem	qusring 13772	If S is a two-sided ideal in R, then U = R / S is a ring, called the quotient ring of R by S. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- U = ( R /.s ( R ~QG S ) )   &    |- I = (2Ideal ` R )   =>    |- ( ( R e. Ring /\ S e. I ) -> U e. Ring )
Theorem	qusmul2 13773	Value of the ring operation in a quotient ring. (Contributed by Thierry Arnoux, 1-Sep-2024.)
|- Q = ( R /.s ( R ~QG I ) )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .X. = ( .r ` Q )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> I e. (2Ideal ` R ) )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( [ X ] ( R ~QG I ) .X. [ Y ] ( R ~QG I ) ) = [ ( X .x. Y ) ] ( R ~QG I ) )
Theorem	crngridl 13774	In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- I = (LIdeal ` R )   &    |- O = (oppr ` R )   =>    |- ( R e. CRing -> I = (LIdeal ` O ) )
Theorem	crng2idl 13775	In a commutative ring, a two-sided ideal is the same as a left ideal. (Contributed by Mario Carneiro, 14-Jun-2015.)
|- I = (LIdeal ` R )   =>    |- ( R e. CRing -> I = (2Ideal ` R ) )
Theorem	qusmulrng 13776	Value of the multiplication operation in a quotient ring of a non-unital ring. Formerly part of proof for quscrng 13777. Similar to qusmul2 13773. (Contributed by Mario Carneiro, 15-Jun-2015.) (Revised by AV, 28-Feb-2025.)
|- .~ = ( R ~QG S )   &    |- H = ( R /.s .~ )   &    |- B = ( Base ` R )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` H )   =>    |- ( ( ( R e. Rng /\ S e. (2Ideal ` R ) /\ S e. (SubGrp ` R ) ) /\ ( X e. B /\ Y e. B ) ) -> ( [ X ] .~ .xb [ Y ] .~ ) = [ ( X .x. Y ) ] .~ )
Theorem	quscrng 13777	The quotient of a commutative ring by an ideal is a commutative ring. (Contributed by Mario Carneiro, 15-Jun-2015.) (Proof shortened by AV, 3-Apr-2025.)
|- U = ( R /.s ( R ~QG S ) )   &    |- I = (LIdeal ` R )   =>    |- ( ( R e. CRing /\ S e. I ) -> U e. CRing )
7.7  The complex numbers as an algebraic extensible structure
7.7.1  Definition and basic properties
Syntax	cpsmet 13778	Extend class notation with the class of all pseudometric spaces.
class PsMet
Syntax	cxmet 13779	Extend class notation with the class of all extended metric spaces.
class *Met
Syntax	cmet 13780	Extend class notation with the class of all metrics.
class Met
Syntax	cbl 13781	Extend class notation with the metric space ball function.
class ball
Syntax	cfbas 13782	Extend class definition to include the class of filter bases.
class fBas
Syntax	cfg 13783	Extend class definition to include the filter generating function.
class filGen
Syntax	cmopn 13784	Extend class notation with a function mapping each metric space to the family of its open sets.
class MetOpen
Syntax	cmetu 13785	Extend class notation with the function mapping metrics to the uniform structure generated by that metric.
class metUnif
Definition	df-psmet 13786*	Define the set of all pseudometrics on a given base set. In a pseudo metric, two distinct points may have a distance zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
|- PsMet = ( x e. _V |-> { d e. ( RR* ^m ( x X. x ) ) | A. y e. x ( ( y d y ) = 0 /\ A. z e. x A. w e. x ( y d z ) <_ ( ( w d y ) +e ( w d z ) ) ) } )
Definition	df-xmet 13787*	Define the set of all extended metrics on a given base set. The definition is similar to df-met 13788, but we also allow the metric to take on the value +oo. (Contributed by Mario Carneiro, 20-Aug-2015.)
|- *Met = ( x e. _V |-> { d e. ( RR* ^m ( x X. x ) ) | A. y e. x A. z e. x ( ( ( y d z ) = 0 <-> y = z ) /\ A. w e. x ( y d z ) <_ ( ( w d y ) +e ( w d z ) ) ) } )
Definition	df-met 13788*	Define the (proper) class of all metrics. (A metric space is the metric's base set paired with the metric. However, we will often also call the metric itself a "metric space".) Equivalent to Definition 14-1.1 of [Gleason] p. 223. (Contributed by NM, 25-Aug-2006.)
|- Met = ( x e. _V |-> { d e. ( RR ^m ( x X. x ) ) | A. y e. x A. z e. x ( ( ( y d z ) = 0 <-> y = z ) /\ A. w e. x ( y d z ) <_ ( ( w d y ) + ( w d z ) ) ) } )
Definition	df-bl 13789*	Define the metric space ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Thierry Arnoux, 11-Feb-2018.)
|- ball = ( d e. _V |-> ( x e. dom dom d , z e. RR* |-> { y e. dom dom d | ( x d y ) < z } ) )
Definition	df-mopn 13790	Define a function whose value is the family of open sets of a metric space. (Contributed by NM, 1-Sep-2006.)
|- MetOpen = ( d e. U. ran *Met |-> ( topGen ` ran ( ball ` d ) ) )
Definition	df-fbas 13791*	Define the class of all filter bases. Note that a filter base on one set is also a filter base for any superset, so there is not a unique base set that can be recovered. (Contributed by Jeff Hankins, 1-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
|- fBas = ( w e. _V |-> { x e. ~P ~P w | ( x =/= (/) /\ (/) e/ x /\ A. y e. x A. z e. x ( x i^i ~P ( y i^i z ) ) =/= (/) ) } )
Definition	df-fg 13792*	Define the filter generating function. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 11-Jul-2015.)
|- filGen = ( w e. _V , x e. ( fBas ` w ) |-> { y e. ~P w | ( x i^i ~P y ) =/= (/) } )
Definition	df-metu 13793*	Define the function mapping metrics to the uniform structure generated by that metric. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
|- metUnif = ( d e. U. ran PsMet |-> ( ( dom dom d X. dom dom d ) filGen ran ( a e. RR+ |-> ( `' d " ( 0 [,) a ) ) ) ) )
Syntax	ccnfld 13794	Extend class notation with the field of complex numbers.
class ℂfld
Definition	df-icnfld 13795	The field of complex numbers. Other number fields and rings can be constructed by applying the ↾s restriction operator. The contract of this set is defined entirely by cnfldex 13797, cnfldadd 13799, cnfldmul 13800, cnfldcj 13801, and cnfldbas 13798. We may add additional members to this in the future. At least for now, this structure does not include a topology, order, a distance function, or function mapping metrics. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Thierry Arnoux, 15-Dec-2017.) (New usage is discouraged.)
|- ℂfld = ( { <. ( Base ` ndx ) , CC >. , <. ( +g ` ndx ) , + >. , <. ( .r ` ndx ) , x. >. } u. { <. ( *r ` ndx ) , * >. } )
Theorem	cnfldstr 13796	The field of complex numbers is a structure. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- ℂfld Struct <. 1 , ; 1 3 >.
Theorem	cnfldex 13797	The field of complex numbers is a set. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 14-Aug-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- ℂfld e. _V
Theorem	cnfldbas 13798	The base set of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- CC = ( Base ` ℂfld )
Theorem	cnfldadd 13799	The addition operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- + = ( +g ` ℂfld )
Theorem	cnfldmul 13800	The multiplication operation of the field of complex numbers. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 6-Oct-2015.) (Revised by Thierry Arnoux, 17-Dec-2017.)
|- x. = ( .r ` ℂfld )