P. 152 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 15101-15200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	aspid 15101	The algebraic span of a subalgebra is itself. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- A = (AlgSpan ` W )   &    |- V = ( Base ` W )   &    |- L = ( LSubSp ` W )   =>    |- ( ( W e. AssAlg /\ S e. (SubRing ` W ) /\ S e. L ) -> ( A ` S ) = S )
Theorem	aspsubrg 15102	The algebraic span of a set of vectors is a subring of the algebra. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- A = (AlgSpan ` W )   &    |- V = ( Base ` W )   =>    |- ( ( W e. AssAlg /\ S C_ V ) -> ( A ` S ) e. (SubRing ` W ) )
Theorem	aspss 15103	Span preserves subset ordering. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- A = (AlgSpan ` W )   &    |- V = ( Base ` W )   =>    |- ( ( W e. AssAlg /\ S C_ V /\ T C_ S ) -> ( A ` T ) C_ ( A ` S ) )
Theorem	aspssid 15104	A set of vectors is a subset of its span. (Contributed by Mario Carneiro, 7-Jan-2015.)
|- A = (AlgSpan ` W )   &    |- V = ( Base ` W )   =>    |- ( ( W e. AssAlg /\ S C_ V ) -> S C_ ( A ` S ) )
Theorem	asclfval 15105*	Function value of the algebra scalar lifting function. (Contributed by Mario Carneiro, 8-Mar-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- .1. = ( 1r ` W )   =>    |- A = ( x e. K |-> ( x .x. .1. ) )
Theorem	asclvald 15106	Value of a mapped algebra scalar. (Contributed by Mario Carneiro, 8-Mar-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- .1. = ( 1r ` W )   &    |- ( ph -> X e. K )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> W e. Ring )   =>    |- ( ph -> ( A ` X ) = ( X .x. .1. ) )
Theorem	asclfnd 15107	Functionality of the algebra scalar lifting function. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> W e. Ring )   =>    |- ( ph -> A Fn K )
Theorem	asclf 15108	The algebra scalar lifting function is a function into the base set. (Contributed by Mario Carneiro, 4-Jul-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. Ring )   &    |- ( ph -> W e. LMod )   &    |- K = ( Base ` F )   &    |- B = ( Base ` W )   =>    |- ( ph -> A : K --> B )
Theorem	asclghm 15109	The algebra scalar lifting function is a group homomorphism. (Contributed by Mario Carneiro, 4-Jul-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. Ring )   &    |- ( ph -> W e. LMod )   =>    |- ( ph -> A e. ( F GrpHom W ) )
Theorem	asclelbas 15110	Lifted scalars are in the base set of the algebra. (Contributed by Zhi Wang, 11-Sep-2025.) (Proof shortened by Thierry Arnoux, 22-Sep-2025.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- ( ph -> W e. AssAlg )   &    |- ( ph -> C e. B )   =>    |- ( ph -> ( A ` C ) e. ( Base ` W ) )
Theorem	ascl0 15111	The scalar 0 embedded into a left module corresponds to the 0 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> W e. Ring )   =>    |- ( ph -> ( A ` ( 0g ` F ) ) = ( 0g ` W ) )
Theorem	ascl1 15112	The scalar 1 embedded into a left module corresponds to the 1 of the left module if the left module is also a ring. (Contributed by AV, 31-Jul-2019.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- ( ph -> W e. LMod )   &    |- ( ph -> W e. Ring )   =>    |- ( ph -> ( A ` ( 1r ` F ) ) = ( 1r ` W ) )
Theorem	asclmul1 15113	Left multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .X. = ( .r ` W )   &    |- .x. = ( .s ` W )   =>    |- ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( ( A ` R ) .X. X ) = ( R .x. X ) )
Theorem	asclmul2 15114	Right multiplication by a lifted scalar is the same as the scalar operation. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- V = ( Base ` W )   &    |- .X. = ( .r ` W )   &    |- .x. = ( .s ` W )   =>    |- ( ( W e. AssAlg /\ R e. K /\ X e. V ) -> ( X .X. ( A ` R ) ) = ( R .x. X ) )
Theorem	ascldimul 15115	The algebra scalar lifting function distributes over multiplication. (Contributed by Mario Carneiro, 8-Mar-2015.) (Proof shortened by SN, 5-Nov-2023.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .X. = ( .r ` W )   &    |- .x. = ( .r ` F )   =>    |- ( ( W e. AssAlg /\ R e. K /\ S e. K ) -> ( A ` ( R .x. S ) ) = ( ( A ` R ) .X. ( A ` S ) ) )
Theorem	asclinvg 15116	The group inverse (negation) of a lifted scalar is the lifted negation of the scalar. (Contributed by AV, 2-Sep-2019.)
|- A = (algSc ` W )   &    |- R = (Scalar ` W )   &    |- B = ( Base ` R )   &    |- I = ( invg ` R )   &    |- J = ( invg ` W )   =>    |- ( ( W e. LMod /\ W e. Ring /\ C e. B ) -> ( J ` ( A ` C ) ) = ( A ` ( I ` C ) ) )
Theorem	asclrhm 15117	The algebra scalar lifting function is a ring homomorphism. (Contributed by Mario Carneiro, 8-Mar-2015.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   =>    |- ( W e. AssAlg -> A e. ( F RingHom W ) )
Theorem	rnascl 15118	The set of lifted scalars is also interpretable as the span of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- A = (algSc ` W )   &    |- .1. = ( 1r ` W )   &    |- N = ( LSpan ` W )   =>    |- ( W e. AssAlg -> ran A = ( N ` { .1. } ) )
Theorem	issubassa2 15119	A subring of a unital algebra is a subspace and thus a subalgebra iff it contains all scalar multiples of the identity. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- A = (algSc ` W )   &    |- L = ( LSubSp ` W )   =>    |- ( ( W e. AssAlg /\ S e. (SubRing ` W ) ) -> ( S e. L <-> ran A C_ S ) )
Theorem	rnasclsubrg 15120	The scalar multiples of the unit vector form a subring of the vectors. (Contributed by SN, 5-Nov-2023.)
|- C = (algSc ` W )   &    |- ( ph -> W e. AssAlg )   =>    |- ( ph -> ran C e. (SubRing ` W ) )
Theorem	rnasclmulcl 15121	(Vector) multiplication is closed for scalar multiples of the unit vector. (Contributed by SN, 5-Nov-2023.)
|- C = (algSc ` W )   &    |- .X. = ( .r ` W )   &    |- ( ph -> W e. AssAlg )   =>    |- ( ( ph /\ ( X e. ran C /\ Y e. ran C ) ) -> ( X .X. Y ) e. ran C )
Theorem	rnasclassa 15122	The scalar multiples of the unit vector form a subalgebra of the vectors. (Contributed by SN, 16-Nov-2023.)
|- A = (algSc ` W )   &    |- U = ( W ↾s ran A )   &    |- ( ph -> W e. AssAlg )   =>    |- ( ph -> U e. AssAlg )
Theorem	ressascl 15123	The lifting of scalars is invariant between subalgebras and superalgebras. (Contributed by Mario Carneiro, 9-Mar-2015.)
|- A = (algSc ` W )   &    |- X = ( W ↾s S )   =>    |- ( S e. (SubRing ` W ) -> A = (algSc ` X ) )
Theorem	asclpropd 15124*	If two structures have the same components (properties), one is an associative algebra iff the other one is. The last hypotheses on 1r can be discharged either by letting W = _V (if strong equality is known on .s) or assuming K is a ring. (Contributed by Mario Carneiro, 5-Jul-2015.)
|- F = (Scalar ` K )   &    |- G = (Scalar ` L )   &    |- ( ph -> P = ( Base ` F ) )   &    |- ( ph -> P = ( Base ` G ) )   &    |- ( ( ph /\ ( x e. P /\ y e. W ) ) -> ( x ( .s ` K ) y ) = ( x ( .s ` L ) y ) )   &    |- ( ph -> ( 1r ` K ) = ( 1r ` L ) )   &    |- ( ph -> ( 1r ` K ) e. W )   =>    |- ( ph -> (algSc ` K ) = (algSc ` L ) )
Theorem	assamulgscmlem1 15125	Lemma 1 for assamulgscm 15127 (induction base). (Contributed by AV, 26-Aug-2019.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- G = (mulGrp ` F )   &    |- .^ = (.g ` G )   &    |- H = (mulGrp ` W )   &    |- E = (.g ` H )   =>    |- ( ( ( A e. B /\ X e. V ) /\ W e. AssAlg ) -> ( 0 E ( A .x. X ) ) = ( ( 0 .^ A ) .x. ( 0 E X ) ) )
Theorem	assamulgscmlem2 15126	Lemma for assamulgscm 15127 (induction step). (Contributed by AV, 26-Aug-2019.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- G = (mulGrp ` F )   &    |- .^ = (.g ` G )   &    |- H = (mulGrp ` W )   &    |- E = (.g ` H )   =>    |- ( y e. NN0 -> ( ( ( A e. B /\ X e. V ) /\ W e. AssAlg ) -> ( ( y E ( A .x. X ) ) = ( ( y .^ A ) .x. ( y E X ) ) -> ( ( y + 1 ) E ( A .x. X ) ) = ( ( ( y + 1 ) .^ A ) .x. ( ( y + 1 ) E X ) ) ) ) )
Theorem	assamulgscm 15127	Exponentiation of a scalar multiplication in an associative algebra: ( a .x. X ) ^ N = ( a ^ N ) .X. ( X ^ N ). (Contributed by AV, 26-Aug-2019.)
|- V = ( Base ` W )   &    |- F = (Scalar ` W )   &    |- B = ( Base ` F )   &    |- .x. = ( .s ` W )   &    |- G = (mulGrp ` F )   &    |- .^ = (.g ` G )   &    |- H = (mulGrp ` W )   &    |- E = (.g ` H )   =>    |- ( ( W e. AssAlg /\ ( N e. NN0 /\ A e. B /\ X e. V ) ) -> ( N E ( A .x. X ) ) = ( ( N .^ A ) .x. ( N E X ) ) )
Theorem	asclmulg 15128	Apply group multiplication to the algebra scalars. (Contributed by Thierry Arnoux, 24-Jul-2024.)
|- A = (algSc ` W )   &    |- F = (Scalar ` W )   &    |- K = ( Base ` F )   &    |- .^ = (.g ` W )   &    |- .* = (.g ` F )   =>    |- ( ( W e. AssAlg /\ N e. NN0 /\ X e. K ) -> ( A ` ( N .* X ) ) = ( N .^ ( A ` X ) ) )
8.2  Abstract multivariate polynomials
8.2.1  Definition and basic properties
Syntax	cmps 15129	Multivariate power series.
class mPwSer
Syntax	cmpl 15130	Multivariate polynomials.
class mPoly
Definition	df-psr 15131*	Define the algebra of power series over the index set i and with coefficients from the ring r. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- mPwSer = ( i e. _V , r e. _V |-> [_ { h e. ( NN0 ^m i ) | ( `' h " NN ) e. Fin } / d ]_ [_ ( ( Base ` r ) ^m d ) / b ]_ ( { <. ( Base ` ndx ) , b >. , <. ( +g ` ndx ) , ( oF ( +g ` r ) |` ( b X. b ) ) >. , <. ( .r ` ndx ) , ( f e. b , g e. b |-> ( k e. d |-> ( r gsumg ( x e. { y e. d | y oR <_ k } |-> ( ( f ` x ) ( .r ` r ) ( g ` ( k oF - x ) ) ) ) ) ) ) >. } u. { <. (Scalar ` ndx ) , r >. , <. ( .s ` ndx ) , ( x e. ( Base ` r ) , f e. b |-> ( ( d X. { x } ) oF ( .r ` r ) f ) ) >. , <. (TopSet ` ndx ) , ( Xt_ ` ( d X. { ( TopOpen ` r ) } ) ) >. } ) )
Definition	df-mplcoe 15132*	Define the subalgebra of the power series algebra generated by the variables; this is the polynomial algebra (the set of power series with finite degree). The index set (which has an element for each variable) is i, the coefficients are in ring r, and for each variable there is a "degree" such that the coefficient is zero for a term where the powers are all greater than those degrees. (Degree is in quotes because there is no guarantee that coefficients below that degree are nonzero, as we do not assume decidable equality for r). (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 7-Oct-2025.)
|- mPoly = ( i e. _V , r e. _V |-> [_ ( i mPwSer r ) / w ]_ ( w ↾s { f e. ( Base ` w ) | E. a e. ( NN0 ^m i ) A. b e. ( NN0 ^m i ) ( A. k e. i ( a ` k ) < ( b ` k ) -> ( f ` b ) = ( 0g ` r ) ) } ) )
Theorem	reldmpsr 15133	The multivariate power series constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- Rel dom mPwSer
Theorem	psrval 15134*	Value of the multivariate power series structure. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- .+ = ( +g ` R )   &    |- .x. = ( .r ` R )   &    |- O = ( TopOpen ` R )   &    |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }   &    |- ( ph -> B = ( K ^m D ) )   &    |- .+b = ( oF .+ |` ( B X. B ) )   &    |- .X. = ( f e. B , g e. B |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( f ` x ) .x. ( g ` ( k oF - x ) ) ) ) ) ) )   &    |- .xb = ( x e. K , f e. B |-> ( ( D X. { x } ) oF .x. f ) )   &    |- ( ph -> J = ( Xt_ ` ( D X. { O } ) ) )   &    |- ( ph -> I e. W )   &    |- ( ph -> R e. X )   =>    |- ( ph -> S = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , .xb >. , <. (TopSet ` ndx ) , J >. } ) )
Theorem	fnpsr 15135	The multivariate power series constructor has a universal domain. (Contributed by Jim Kingdon, 16-Jun-2025.)
|- mPwSer Fn ( _V X. _V )
Theorem	psrvalstrd 15136	The multivariate power series structure is a function. (Contributed by Mario Carneiro, 8-Feb-2015.)
|- ( ph -> B e. X )   &    |- ( ph -> .+ e. Y )   &    |- ( ph -> .X. e. Z )   &    |- ( ph -> R e. W )   &    |- ( ph -> .x. e. P )   &    |- ( ph -> J e. Q )   =>    |- ( ph -> ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .X. >. } u. { <. (Scalar ` ndx ) , R >. , <. ( .s ` ndx ) , .x. >. , <. (TopSet ` ndx ) , J >. } ) Struct <. 1 , 9 >. )
Theorem	psrbag 15137*	Elementhood in the set of finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
Theorem	psrbagf 15138*	A finite bag is a function. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 30-Jul-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( F e. D -> F : I --> NN0 )
Theorem	psrbagfsupp 15139*	Finite bags have finite support. (Contributed by Stefan O'Rear, 9-Mar-2015.) (Revised by AV, 18-Jul-2019.) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( F e. D -> F finSupp 0 )
Theorem	fczpsrbag 15140*	The constant function equal to zero is a finite bag. (Contributed by AV, 8-Jul-2019.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. V -> ( x e. I |-> 0 ) e. D )
Theorem	psrbaglesuppg 15141*	The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' G " NN ) C_ ( `' F " NN ) )
Theorem	psrbaglesupp 15142*	The support of a dominated bag is smaller than the dominating bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( `' G " NN ) C_ ( `' F " NN ) )
Theorem	psrbagfi 15143*	A finite index set gives a simpler expression for finite bags. (Contributed by Jim Kingdon, 23-Nov-2025.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( I e. Fin -> D = ( NN0 ^m I ) )
Theorem	psrbaglecl 15144*	The set of finite bags is downward-closed. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> G e. D )
Theorem	psrbagaddclfi 15145*	The sum of two finite bags is a finite bag. (Contributed by Mario Carneiro, 9-Jan-2015.) Shorten proof and remove a sethood antecedent. (Revised by SN, 7-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G e. D /\ I e. Fin ) -> ( F oF + G ) e. D )
Theorem	psrbagcon 15146*	The analogue of the statement " 0 <_ G <_ F implies 0 <_ F - G <_ F " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 5-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) -> ( ( F oF - G ) e. D /\ ( F oF - G ) oR <_ F ) )
Theorem	psrbaglefifi 15147*	There are finitely many bags dominated by a given bag. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Mario Carneiro, 25-Jan-2015.) (Revised by Jim Kingdon, 28-Jul-2026.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ( F e. D /\ I e. Fin ) -> { y e. D | y oR <_ F } e. Fin )
Theorem	psrbagconcl 15148*	The complement of a bag is a bag. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- S = { y e. D | y oR <_ F }   =>    |- ( ( F e. D /\ X e. S ) -> ( F oF - X ) e. S )
Theorem	psrbagconf1o 15149*	Bag complementation is a bijection on the set of bags dominated by a given bag F. (Contributed by Mario Carneiro, 29-Dec-2014.) Remove a sethood antecedent. (Revised by SN, 6-Aug-2024.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- S = { y e. D | y oR <_ F }   =>    |- ( F e. D -> ( x e. S |-> ( F oF - x ) ) : S -1-1-onto-> S )
Theorem	psrbasg 15150*	The base set of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 8-Jul-2019.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- B = ( Base ` S )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. W )   =>    |- ( ph -> B = ( K ^m D ) )
Theorem	psrelbas 15151*	An element of the set of power series is a function on the coefficients. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : D --> K )
Theorem	psrelbasfi 15152	Simpler form of psrelbas 15151 when the index set is finite. (Contributed by Jim Kingdon, 27-Nov-2025.)
|- S = ( I mPwSer R )   &    |- K = ( Base ` R )   &    |- ( ph -> I e. Fin )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : ( NN0 ^m I ) --> K )
Theorem	psrelbasfun 15153	An element of the set of power series is a function. (Contributed by AV, 17-Jul-2019.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   =>    |- ( X e. B -> Fun X )
Theorem	psrplusgg 15154	The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` S )   =>    |- ( ( I e. V /\ R e. W ) -> .+b = ( oF .+ |` ( B X. B ) ) )
Theorem	psradd 15155	The addition operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` S )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )
Theorem	psraddcl 15156	Closure of the power series addition operation. (Contributed by Mario Carneiro, 28-Dec-2014.) Generalize to magmas. (Revised by SN, 12-Apr-2025.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- ( ph -> R e. Mgm )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+ Y ) e. B )
Theorem	rhmpsrfilem2 15157*	Lemma for rhmpsr et al. (Contributed by SN, 8-Feb-2025.)
|- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- ( ph -> R e. Ring )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> X : D --> ( Base ` R ) )   &    |- ( ph -> Y : D --> ( Base ` R ) )   =>    |- ( ( ph /\ k e. D ) -> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( X ` x ) ( .r ` R ) ( Y ` ( k oF - x ) ) ) ) ) e. ( Base ` R ) )
Theorem	psrmulrg 15158*	The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.) (Proof shortened by AV, 2-Mar-2024.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` S )   &    |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }   =>    |- ( ( I e. V /\ R e. W ) -> .xb = ( f e. B , g e. B |-> ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( f ` x ) .x. ( g ` ( k oF - x ) ) ) ) ) ) ) )
Theorem	psrmulfval 15159*	The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` S )   &    |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   =>    |- ( ph -> ( F .xb G ) = ( k e. D |-> ( R gsumg ( x e. { y e. D | y oR <_ k } |-> ( ( F ` x ) .x. ( G ` ( k oF - x ) ) ) ) ) ) )
Theorem	psrmulvalfi 15160*	The multiplication operation of the multivariate power series structure. (Contributed by Mario Carneiro, 28-Dec-2014.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .x. = ( .r ` R )   &    |- .xb = ( .r ` S )   &    |- D = { h e. ( NN0 ^m I ) | ( `' h " NN ) e. Fin }   &    |- ( ph -> F e. B )   &    |- ( ph -> G e. B )   &    |- ( ph -> X e. D )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Ring )   =>    |- ( ph -> ( ( F .xb G ) ` X ) = ( R gsumg ( k e. { y e. D | y oR <_ X } |-> ( ( F ` k ) .x. ( G ` ( X oF - k ) ) ) ) ) )
Theorem	psrmulclfilem 15161*	Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .x. = ( .r ` S )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   =>    |- ( ph -> ( X .x. Y ) e. B )
Theorem	psrmulclfi 15162	Closure of the power series multiplication operation. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .x. = ( .r ` S )   &    |- ( ph -> R e. Ring )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .x. Y ) e. B )
Theorem	psr0cl 15163*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- B = ( Base ` S )   =>    |- ( ph -> ( D X. { .0. } ) e. B )
Theorem	psr0lid 15164*	The zero element of the ring of power series is a left identity. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- B = ( Base ` S )   &    |- .+ = ( +g ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( ( D X. { .0. } ) .+ X ) = X )
Theorem	psrnegcl 15165*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( N o. X ) e. B )
Theorem	psrlinv 15166*	The negative function in the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- ( ph -> X e. B )   &    |- .0. = ( 0g ` R )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( ( N o. X ) .+ X ) = ( D X. { .0. } ) )
Theorem	psrgrp 15167	The ring of power series is a group. (Contributed by Mario Carneiro, 29-Dec-2014.) (Proof shortened by SN, 7-Feb-2025.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> S e. Grp )
Theorem	psr0 15168*	The zero element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- O = ( 0g ` R )   &    |- .0. = ( 0g ` S )   =>    |- ( ph -> .0. = ( D X. { O } ) )
Theorem	psrneg 15169*	The negative function of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. V )   &    |- ( ph -> R e. Grp )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- N = ( invg ` R )   &    |- B = ( Base ` S )   &    |- M = ( invg ` S )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
Theorem	psr1clfi 15170*	The identity element of the ring of power series. (Contributed by Mario Carneiro, 29-Dec-2014.)
|- S = ( I mPwSer R )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Ring )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- .0. = ( 0g ` R )   &    |- .1. = ( 1r ` R )   &    |- U = ( x e. D |-> if ( x = ( I X. { 0 } ) , .1. , .0. ) )   &    |- B = ( Base ` S )   =>    |- ( ph -> U e. B )
Theorem	reldmmpl 15171	The multivariate polynomial constructor is a proper binary operator. (Contributed by Mario Carneiro, 21-Mar-2015.)
|- Rel dom mPoly
Theorem	mplvalcoe 15172*	Value of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) }   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
Theorem	mplbascoe 15173*	Base set of the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> U = { f e. B | E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( f ` b ) = .0. ) } )
Theorem	mplelbascoe 15174*	Property of being a polynomial. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 25-Jun-2019.) (Revised by Jim Kingdon, 4-Nov-2025.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- B = ( Base ` S )   &    |- .0. = ( 0g ` R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> ( X e. U <-> ( X e. B /\ E. a e. ( NN0 ^m I ) A. b e. ( NN0 ^m I ) ( A. k e. I ( a ` k ) < ( b ` k ) -> ( X ` b ) = .0. ) ) ) )
Theorem	fnmpl 15175	mPoly has universal domain. (Contributed by Jim Kingdon, 5-Nov-2025.)
|- mPoly Fn ( _V X. _V )
Theorem	mplrcl 15176	Reverse closure for the polynomial index set. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 30-Aug-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   =>    |- ( X e. B -> I e. _V )
Theorem	mplval2g 15177	Self-referential expression for the set of multivariate polynomials. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- U = ( Base ` P )   =>    |- ( ( I e. V /\ R e. W ) -> P = ( S ↾s U ) )
Theorem	mplbasss 15178	The set of polynomials is a subset of the set of power series. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- U = ( Base ` P )   &    |- B = ( Base ` S )   =>    |- U C_ B
Theorem	mplelf 15179*	A polynomial is defined as a function on the coefficients. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- K = ( Base ` R )   &    |- B = ( Base ` P )   &    |- D = { f e. ( NN0 ^m I ) | ( `' f " NN ) e. Fin }   &    |- ( ph -> X e. B )   =>    |- ( ph -> X : D --> K )
Theorem	mplsubgfilemm 15180*	Lemma for mplsubgfi 15183. There exists a polynomial. (Contributed by Jim Kingdon, 21-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> E. j j e. U )
Theorem	mplsubgfilemcl 15181	Lemma for mplsubgfi 15183. The sum of two polynomials is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- ( ph -> Y e. U )   &    |- .+ = ( +g ` S )   =>    |- ( ph -> ( X .+ Y ) e. U )
Theorem	mplsubgfileminv 15182	Lemma for mplsubgfi 15183. The additive inverse of a polynomial is a polynomial. (Contributed by Jim Kingdon, 26-Nov-2025.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. U )   &    |- N = ( invg ` S )   =>    |- ( ph -> ( N ` X ) e. U )
Theorem	mplsubgfi 15183	The set of polynomials is closed under addition, i.e. it is a subgroup of the set of power series. (Contributed by Mario Carneiro, 8-Jan-2015.) (Proof shortened by AV, 16-Jul-2019.)
|- S = ( I mPwSer R )   &    |- P = ( I mPoly R )   &    |- U = ( Base ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> U e. (SubGrp ` S ) )
Theorem	mpl0fi 15184*	The zero polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   &    |- O = ( 0g ` R )   &    |- .0. = ( 0g ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   =>    |- ( ph -> .0. = ( x e. ( NN0 ^m I ) |-> O ) )
Theorem	mplplusgg 15185	Value of addition in a polynomial ring. (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- Y = ( I mPoly R )   &    |- S = ( I mPwSer R )   &    |- .+ = ( +g ` Y )   =>    |- ( ( I e. V /\ R e. W ) -> .+ = ( +g ` S ) )
Theorem	mpladd 15186	The addition operation on multivariate polynomials. (Contributed by Mario Carneiro, 9-Jan-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- .+ = ( +g ` R )   &    |- .+b = ( +g ` P )   &    |- ( ph -> X e. B )   &    |- ( ph -> Y e. B )   =>    |- ( ph -> ( X .+b Y ) = ( X oF .+ Y ) )
Theorem	mplnegfi 15187	The negative function on multivariate polynomials. (Contributed by SN, 25-May-2024.)
|- P = ( I mPoly R )   &    |- B = ( Base ` P )   &    |- N = ( invg ` R )   &    |- M = ( invg ` P )   &    |- ( ph -> I e. Fin )   &    |- ( ph -> R e. Grp )   &    |- ( ph -> X e. B )   =>    |- ( ph -> ( M ` X ) = ( N o. X ) )
Theorem	mplgrpfi 15188	The polynomial ring is a group. (Contributed by Mario Carneiro, 9-Jan-2015.)
|- P = ( I mPoly R )   =>    |- ( ( I e. Fin /\ R e. Grp ) -> P e. Grp )
PART 9  BASIC TOPOLOGY
9.1  Topology
9.1.1  Topological spaces A topology on a set is a set of subsets of that set, called open sets, which satisfy certain conditions. One condition is that the whole set be an open set. Therefore, a set is recoverable from a topology on it (as its union), and it may sometimes be more convenient to consider topologies without reference to the underlying set.
9.1.1.1  Topologies
Syntax	ctop 15189	Syntax for the class of topologies.
class Top
Definition	df-top 15190*	Define the class of topologies. It is a proper class. See istopg 15191 and istopfin 15192 for the corresponding characterizations, using respectively binary intersections like in this definition and nonempty finite intersections. The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)
|- Top = { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }
Theorem	istopg 15191*	Express the predicate " J is a topology". See istopfin 15192 for another characterization using nonempty finite intersections instead of binary intersections. Note: In the literature, a topology is often represented by a calligraphic letter T, which resembles the letter J. This confusion may have led to J being used by some authors (e.g., K. D. Joshi, Introduction to General Topology (1983), p. 114) and it is convenient for us since we later use T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
|- ( J e. A -> ( J e. Top <-> ( A. x ( x C_ J -> U. x e. J ) /\ A. x e. J A. y e. J ( x i^i y ) e. J ) ) )
Theorem	istopfin 15192*	Express the predicate " J is a topology" using nonempty finite intersections instead of binary intersections as in istopg 15191. It is not clear we can prove the converse without adding additional conditions. (Contributed by NM, 19-Jul-2006.) (Revised by Jim Kingdon, 14-Jan-2023.)
|- ( J e. Top -> ( A. x ( x C_ J -> U. x e. J ) /\ A. x ( ( x C_ J /\ x =/= (/) /\ x e. Fin ) -> |^| x e. J ) ) )
Theorem	uniopn 15193	The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( ( J e. Top /\ A C_ J ) -> U. A e. J )
Theorem	iunopn 15194*	The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
|- ( ( J e. Top /\ A. x e. A B e. J ) -> U_ x e. A B e. J )
Theorem	inopn 15195	The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A i^i B ) e. J )
Theorem	fiinopn 15196	The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
|- ( J e. Top -> ( ( A C_ J /\ A =/= (/) /\ A e. Fin ) -> |^| A e. J ) )
Theorem	unopn 15197	The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
|- ( ( J e. Top /\ A e. J /\ B e. J ) -> ( A u. B ) e. J )
Theorem	0opn 15198	The empty set is an open subset of any topology. (Contributed by Stefan Allan, 27-Feb-2006.)
|- ( J e. Top -> (/) e. J )
Theorem	0ntop 15199	The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
|- -. (/) e. Top
Theorem	topopn 15200	The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
|- X = U. J   =>    |- ( J e. Top -> X e. J )