P. 211 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cnextfun 21001	If the target space is Hausdorff, a continuous extension is a function (Contributed by Thierry Arnoux, 20-Dec-2017.)
|- C = U. J   &    |- B = U. K   =>    |- ( ( ( J e. Top /\ K e. Haus ) /\ ( F : A --> B /\ A C_ C ) ) -> Fun ( ( JCnExt K ) ` F ) )
Theorem	cnextfvval 21002*	The value of the continuous extension of a given function F at a point X. (Contributed by Thierry Arnoux, 21-Dec-2017.)
|- C = U. J   &    |- B = U. K   &    |- ( ph -> J e. Top )   &    |- ( ph -> K e. Haus )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A C_ C )   &    |- ( ph -> ( ( cls ` J ) ` A ) = C )   &    |- ( ( ph /\ x e. C ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) =/= (/) )   =>    |- ( ( ph /\ X e. C ) -> ( ( ( JCnExt K ) ` F ) ` X ) = U. ( ( K fLimf ( ( ( nei ` J ) ` { X } ) ↾t A ) ) ` F ) )
Theorem	cnextf 21003*	Extension by continuity. The extension by continuity is a function. (Contributed by Thierry Arnoux, 25-Dec-2017.)
|- C = U. J   &    |- B = U. K   &    |- ( ph -> J e. Top )   &    |- ( ph -> K e. Haus )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A C_ C )   &    |- ( ph -> ( ( cls ` J ) ` A ) = C )   &    |- ( ( ph /\ x e. C ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) =/= (/) )   =>    |- ( ph -> ( ( JCnExt K ) ` F ) : C --> B )
Theorem	cnextcn 21004*	Extension by continuity. Theorem 1 of [BourbakiTop1] p. I.57. Given a topology J on C, a subset A dense in C, this states a condition for F from A to a regular space K to be extensible by continuity (Contributed by Thierry Arnoux, 1-Jan-2018.)
|- C = U. J   &    |- B = U. K   &    |- ( ph -> J e. Top )   &    |- ( ph -> K e. Haus )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A C_ C )   &    |- ( ph -> ( ( cls ` J ) ` A ) = C )   &    |- ( ( ph /\ x e. C ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) =/= (/) )   &    |- ( ph -> K e. Reg )   =>    |- ( ph -> ( ( JCnExt K ) ` F ) e. ( J Cn K ) )
Theorem	cnextfres1 21005*	F and its extension by continuity agree on the domain of F. (Contributed by Thierry Arnoux, 17-Jan-2018.)
|- C = U. J   &    |- B = U. K   &    |- ( ph -> J e. Top )   &    |- ( ph -> K e. Haus )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> A C_ C )   &    |- ( ph -> ( ( cls ` J ) ` A ) = C )   &    |- ( ( ph /\ x e. C ) -> ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) =/= (/) )   &    |- ( ph -> K e. Reg )   &    |- ( ph -> F e. ( ( J ↾t A ) Cn K ) )   =>    |- ( ph -> ( ( ( JCnExt K ) ` F ) |` A ) = F )
Theorem	cnextfres 21006	F and its extension by continuity agree on the domain of F. (Contributed by Thierry Arnoux, 29-Aug-2020.)
|- C = U. J   &    |- B = U. K   &    |- ( ph -> J e. Top )   &    |- ( ph -> K e. Haus )   &    |- ( ph -> A C_ C )   &    |- ( ph -> F e. ( ( J ↾t A ) Cn K ) )   &    |- ( ph -> X e. A )   =>    |- ( ph -> ( ( ( JCnExt K ) ` F ) ` X ) = ( F ` X ) )
12.2.6  Topological groups
Syntax	ctmd 21007	Extend class notation with the class of all topological monoids.
class TopMnd
Syntax	ctgp 21008	Extend class notation with the class of all topological groups.
class TopGrp
Definition	df-tmd 21009*	Define the class of all topological monoids. A topological monoid is a monoid whose operation is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- TopMnd = { f e. ( Mnd i^i TopSp ) | [. ( TopOpen ` f ) / j ]. ( +f ` f ) e. ( ( j tX j ) Cn j ) }
Definition	df-tgp 21010*	Define the class of all topological groups. A topological group is a group whose operation and inverse function are continuous. (Contributed by FL, 18-Apr-2010.)
|- TopGrp = { f e. ( Grp i^i TopMnd ) | [. ( TopOpen ` f ) / j ]. ( invg ` f ) e. ( j Cn j ) }
Theorem	istmd 21011	The predicate "is a topological monoid". (Contributed by Mario Carneiro, 19-Sep-2015.)
|- F = ( +f ` G )   &    |- J = ( TopOpen ` G )   =>    |- ( G e. TopMnd <-> ( G e. Mnd /\ G e. TopSp /\ F e. ( ( J tX J ) Cn J ) ) )
Theorem	tmdmnd 21012	A topological monoid is a monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- ( G e. TopMnd -> G e. Mnd )
Theorem	tmdtps 21013	A topological monoid is a topological space. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- ( G e. TopMnd -> G e. TopSp )
Theorem	istgp 21014	The predicate "is a topological group". Definition of [BourbakiTop1] p. III.1 (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- J = ( TopOpen ` G )   &    |- I = ( invg ` G )   =>    |- ( G e. TopGrp <-> ( G e. Grp /\ G e. TopMnd /\ I e. ( J Cn J ) ) )
Theorem	tgpgrp 21015	A topological group is a group. (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( G e. TopGrp -> G e. Grp )
Theorem	tgptmd 21016	A topological group is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- ( G e. TopGrp -> G e. TopMnd )
Theorem	tgptps 21017	A topological group is a topological space. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- ( G e. TopGrp -> G e. TopSp )
Theorem	tmdtopon 21018	The topology of a topological monoid. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- J = ( TopOpen ` G )   &    |- X = ( Base ` G )   =>    |- ( G e. TopMnd -> J e. (TopOn ` X ) )
Theorem	tgptopon 21019	The topology of a topological group. (Contributed by Mario Carneiro, 27-Jun-2014.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- J = ( TopOpen ` G )   &    |- X = ( Base ` G )   =>    |- ( G e. TopGrp -> J e. (TopOn ` X ) )
Theorem	tmdcn 21020	In a topological monoid, the operation F representing the functionalization of the operator slot +g is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- J = ( TopOpen ` G )   &    |- F = ( +f ` G )   =>    |- ( G e. TopMnd -> F e. ( ( J tX J ) Cn J ) )
Theorem	tgpcn 21021	In a topological group, the operation F representing the functionalization of the operator slot +g is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 13-Aug-2015.)
|- J = ( TopOpen ` G )   &    |- F = ( +f ` G )   =>    |- ( G e. TopGrp -> F e. ( ( J tX J ) Cn J ) )
Theorem	tgpinv 21022	In a topological group, the inverse function is continuous. (Contributed by FL, 21-Jun-2010.) (Revised by FL, 27-Jun-2014.)
|- J = ( TopOpen ` G )   &    |- I = ( invg ` G )   =>    |- ( G e. TopGrp -> I e. ( J Cn J ) )
Theorem	grpinvhmeo 21023	The inverse function in a topological group is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- J = ( TopOpen ` G )   &    |- I = ( invg ` G )   =>    |- ( G e. TopGrp -> I e. ( J Homeo J ) )
Theorem	cnmpt1plusg 21024*	Continuity of the group sum; analogue of cnmpt12f 20603 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- J = ( TopOpen ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. TopMnd )   &    |- ( ph -> K e. (TopOn ` X ) )   &    |- ( ph -> ( x e. X |-> A ) e. ( K Cn J ) )   &    |- ( ph -> ( x e. X |-> B ) e. ( K Cn J ) )   =>    |- ( ph -> ( x e. X |-> ( A .+ B ) ) e. ( K Cn J ) )
Theorem	cnmpt2plusg 21025*	Continuity of the group sum; analogue of cnmpt22f 20612 which cannot be used directly because +g is not a function. (Contributed by Mario Carneiro, 23-Aug-2015.)
|- J = ( TopOpen ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. TopMnd )   &    |- ( ph -> K e. (TopOn ` X ) )   &    |- ( ph -> L e. (TopOn ` Y ) )   &    |- ( ph -> ( x e. X , y e. Y |-> A ) e. ( ( K tX L ) Cn J ) )   &    |- ( ph -> ( x e. X , y e. Y |-> B ) e. ( ( K tX L ) Cn J ) )   =>    |- ( ph -> ( x e. X , y e. Y |-> ( A .+ B ) ) e. ( ( K tX L ) Cn J ) )
Theorem	tmdcn2 21026*	Write out the definition of continuity of +g explicitly. (Contributed by Mario Carneiro, 20-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- .+ = ( +g ` G )   =>    |- ( ( ( G e. TopMnd /\ U e. J ) /\ ( X e. B /\ Y e. B /\ ( X .+ Y ) e. U ) ) -> E. u e. J E. v e. J ( X e. u /\ Y e. v /\ A. x e. u A. y e. v ( x .+ y ) e. U ) )
Theorem	tgpsubcn 21027	In a topological group, the "subtraction" (or "division") is continuous. Axiom GT' of [BourbakiTop1] p. III.1 (Contributed by FL, 21-Jun-2010.) (Revised by Mario Carneiro, 19-Mar-2015.)
|- J = ( TopOpen ` G )   &    |- .- = ( -g ` G )   =>    |- ( G e. TopGrp -> .- e. ( ( J tX J ) Cn J ) )
Theorem	istgp2 21028	A group with a topology is a topological group iff the subtraction operation is continuous. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- J = ( TopOpen ` G )   &    |- .- = ( -g ` G )   =>    |- ( G e. TopGrp <-> ( G e. Grp /\ G e. TopSp /\ .- e. ( ( J tX J ) Cn J ) ) )
Theorem	tmdmulg 21029*	In a topological monoid, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- J = ( TopOpen ` G )   &    |- .x. = (.g ` G )   &    |- B = ( Base ` G )   =>    |- ( ( G e. TopMnd /\ N e. NN0 ) -> ( x e. B |-> ( N .x. x ) ) e. ( J Cn J ) )
Theorem	tgpmulg 21030*	In a topological group, the n-times group multiple function is continuous. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- J = ( TopOpen ` G )   &    |- .x. = (.g ` G )   &    |- B = ( Base ` G )   =>    |- ( ( G e. TopGrp /\ N e. ZZ ) -> ( x e. B |-> ( N .x. x ) ) e. ( J Cn J ) )
Theorem	tgpmulg2 21031	In a topological monoid, the group multiple function is jointly continuous (although this is not saying much as one of the factors is discrete). Use zdis 21736 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- J = ( TopOpen ` G )   &    |- .x. = (.g ` G )   =>    |- ( G e. TopGrp -> .x. e. ( ( ~P ZZ tX J ) Cn J ) )
Theorem	tmdgsum 21032*	In a topological monoid, the group sum operation is a continuous function from the function space to the base topology. This theorem is not true when A is infinite, because in this case for any basic open set of the domain one of the factors will be the whole space, so by varying the value of the functions to sum at this index, one can achieve any desired sum. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
|- J = ( TopOpen ` G )   &    |- B = ( Base ` G )   =>    |- ( ( G e. CMnd /\ G e. TopMnd /\ A e. Fin ) -> ( x e. ( B ^m A ) |-> ( G gsumg x ) ) e. ( ( J ^ko ~P A ) Cn J ) )
Theorem	tmdgsum2 21033*	For any neighborhood U of n X, there is a neighborhood u of X such that any sum of n elements in u sums to an element of U. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- J = ( TopOpen ` G )   &    |- B = ( Base ` G )   &    |- .x. = (.g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopMnd )   &    |- ( ph -> A e. Fin )   &    |- ( ph -> U e. J )   &    |- ( ph -> X e. B )   &    |- ( ph -> ( ( # ` A ) .x. X ) e. U )   =>    |- ( ph -> E. u e. J ( X e. u /\ A. f e. ( u ^m A ) ( G gsumg f ) e. U ) )
Theorem	oppgtmd 21034	The opposite of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- O = (oppg ` G )   =>    |- ( G e. TopMnd -> O e. TopMnd )
Theorem	oppgtgp 21035	The opposite of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- O = (oppg ` G )   =>    |- ( G e. TopGrp -> O e. TopGrp )
Theorem	distgp 21036	Any group equipped with the discrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   =>    |- ( ( G e. Grp /\ J = ~P B ) -> G e. TopGrp )
Theorem	indistgp 21037	Any group equipped with the indiscrete topology is a topological group. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   =>    |- ( ( G e. Grp /\ J = { (/) , B } ) -> G e. TopGrp )
Theorem	symgtgp 21038	The symmetric group is a topological group. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- G = ( SymGrp ` A )   =>    |- ( A e. V -> G e. TopGrp )
Theorem	tmdlactcn 21039*	The left group action of element A in a topological monoid G is a continuous function. (Contributed by FL, 18-Mar-2008.) (Revised by Mario Carneiro, 14-Aug-2015.)
|- F = ( x e. X |-> ( A .+ x ) )   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- J = ( TopOpen ` G )   =>    |- ( ( G e. TopMnd /\ A e. X ) -> F e. ( J Cn J ) )
Theorem	tgplacthmeo 21040*	The left group action of element A in a topological group G is a homeomorphism from the group to itself. (Contributed by Mario Carneiro, 14-Aug-2015.)
|- F = ( x e. X |-> ( A .+ x ) )   &    |- X = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- J = ( TopOpen ` G )   =>    |- ( ( G e. TopGrp /\ A e. X ) -> F e. ( J Homeo J ) )
Theorem	submtmd 21041	A submonoid of a topological monoid is a topological monoid. (Contributed by Mario Carneiro, 6-Oct-2015.)
|- H = ( G ↾s S )   =>    |- ( ( G e. TopMnd /\ S e. (SubMnd ` G ) ) -> H e. TopMnd )
Theorem	subgtgp 21042	A subgroup of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- H = ( G ↾s S )   =>    |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> H e. TopGrp )
Theorem	subgntr 21043	A subgroup of a topological group with nonempty interior is open. Alternatively, dual to clssubg 21045, the interior of a subgroup is either a subgroup, or empty. (Contributed by Mario Carneiro, 19-Sep-2015.)
|- J = ( TopOpen ` G )   =>    |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) /\ A e. ( ( int ` J ) ` S ) ) -> S e. J )
Theorem	opnsubg 21044	An open subgroup of a topological group is also closed. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- J = ( TopOpen ` G )   =>    |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) /\ S e. J ) -> S e. ( Clsd ` J ) )
Theorem	clssubg 21045	The closure of a subgroup in a topological group is a subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- J = ( TopOpen ` G )   =>    |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) ) -> ( ( cls ` J ) ` S ) e. (SubGrp ` G ) )
Theorem	clsnsg 21046	The closure of a normal subgroup is a normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- J = ( TopOpen ` G )   =>    |- ( ( G e. TopGrp /\ S e. (NrmSGrp ` G ) ) -> ( ( cls ` J ) ` S ) e. (NrmSGrp ` G ) )
Theorem	cldsubg 21047	A subgroup of finite index is closed iff it is open. (Contributed by Mario Carneiro, 20-Sep-2015.)
|- J = ( TopOpen ` G )   &    |- R = ( G ~QG S )   &    |- X = ( Base ` G )   =>    |- ( ( G e. TopGrp /\ S e. (SubGrp ` G ) /\ ( X /. R ) e. Fin ) -> ( S e. ( Clsd ` J ) <-> S e. J ) )
Theorem	tgpconcompeqg 21048*	The connected component containing A is the left coset of the identity component containing A. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- X = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- J = ( TopOpen ` G )   &    |- S = U. { x e. ~P X | ( .0. e. x /\ ( J ↾t x ) e. Con ) }   &    |- .~ = ( G ~QG S )   =>    |- ( ( G e. TopGrp /\ A e. X ) -> [ A ] .~ = U. { x e. ~P X | ( A e. x /\ ( J ↾t x ) e. Con ) } )
Theorem	tgpconcomp 21049*	The identity component, the connected component containing the identity element, is a closed (concompcld 20371) normal subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- X = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- J = ( TopOpen ` G )   &    |- S = U. { x e. ~P X | ( .0. e. x /\ ( J ↾t x ) e. Con ) }   =>    |- ( G e. TopGrp -> S e. (NrmSGrp ` G ) )
Theorem	tgpconcompss 21050*	The identity component is a subset of any open subgroup. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- X = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- J = ( TopOpen ` G )   &    |- S = U. { x e. ~P X | ( .0. e. x /\ ( J ↾t x ) e. Con ) }   =>    |- ( ( G e. TopGrp /\ T e. (SubGrp ` G ) /\ T e. J ) -> S C_ T )
Theorem	ghmcnp 21051	A group homomorphism on topological groups is continuous everywhere if it is continuous at any point. (Contributed by Mario Carneiro, 21-Oct-2015.)
|- X = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- K = ( TopOpen ` H )   =>    |- ( ( G e. TopMnd /\ H e. TopMnd /\ F e. ( G GrpHom H ) ) -> ( F e. ( ( J CnP K ) ` A ) <-> ( A e. X /\ F e. ( J Cn K ) ) ) )
Theorem	snclseqg 21052	The coset of the closure of the identity is the closure of a point. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- X = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- .0. = ( 0g ` G )   &    |- .~ = ( G ~QG S )   &    |- S = ( ( cls ` J ) ` { .0. } )   =>    |- ( ( G e. TopGrp /\ A e. X ) -> [ A ] .~ = ( ( cls ` J ) ` { A } ) )
Theorem	tgphaus 21053	A topological group is Hausdorff iff the identity subgroup is closed. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- .0. = ( 0g ` G )   &    |- J = ( TopOpen ` G )   =>    |- ( G e. TopGrp -> ( J e. Haus <-> { .0. } e. ( Clsd ` J ) ) )
Theorem	tgpt1 21054	Hausdorff and T1 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- J = ( TopOpen ` G )   =>    |- ( G e. TopGrp -> ( J e. Haus <-> J e. Fre ) )
Theorem	tgpt0 21055	Hausdorff and T0 are equivalent for topological groups. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- J = ( TopOpen ` G )   =>    |- ( G e. TopGrp -> ( J e. Haus <-> J e. Kol2 ) )
Theorem	qustgpopn 21056*	A quotient map in a topological group is an open map. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- H = ( G /.s ( G ~QG Y ) )   &    |- X = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- K = ( TopOpen ` H )   &    |- F = ( x e. X |-> [ x ] ( G ~QG Y ) )   =>    |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ S e. J ) -> ( F " S ) e. K )
Theorem	qustgplem 21057*	Lemma for qustgp 21058. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- H = ( G /.s ( G ~QG Y ) )   &    |- X = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- K = ( TopOpen ` H )   &    |- F = ( x e. X |-> [ x ] ( G ~QG Y ) )   &    |- .- = ( z e. X , w e. X |-> [ ( z ( -g ` G ) w ) ] ( G ~QG Y ) )   =>    |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) ) -> H e. TopGrp )
Theorem	qustgp 21058	The quotient of a topological group is a topological group. (Contributed by Mario Carneiro, 17-Sep-2015.)
|- H = ( G /.s ( G ~QG Y ) )   =>    |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) ) -> H e. TopGrp )
Theorem	qustgphaus 21059	The quotient of a topological group by a closed normal subgroup is a Hausdorff topological group. In particular, the quotient by the closure of the identity is a Hausdorff topological group, isomorphic to both the Kolmogorov quotient and the Hausdorff quotient operations on topological spaces (because T0 and Hausdorff coincide for topological groups). (Contributed by Mario Carneiro, 22-Sep-2015.)
|- H = ( G /.s ( G ~QG Y ) )   &    |- J = ( TopOpen ` G )   &    |- K = ( TopOpen ` H )   =>    |- ( ( G e. TopGrp /\ Y e. (NrmSGrp ` G ) /\ Y e. ( Clsd ` J ) ) -> K e. Haus )
Theorem	prdstmdd 21060	The product of a family of topological monoids is a topological monoid. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I -->TopMnd )   =>    |- ( ph -> Y e. TopMnd )
Theorem	prdstgpd 21061	The product of a family of topological groups is a topological group. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- Y = ( S X_s R )   &    |- ( ph -> I e. W )   &    |- ( ph -> S e. V )   &    |- ( ph -> R : I --> TopGrp )   =>    |- ( ph -> Y e. TopGrp )
12.2.7  Infinite group sum on topological groups
Syntax	ctsu 21062	Extend class notation to include infinite group sums in a topological group.
class tsums
Definition	df-tsms 21063*	Define the set of limit points of an infinite group sum for the topological group G. If G is Hausdorff, then there will be at most one element in this set and U. ( W tsums F ) selects this unique element if it exists. ( W tsums F ) ~~ 1o is a way to say that the sum exists and is unique. Note that unlike sum_ (df-sum 13731) and gsumg (df-gsum 15291), this does not return the sum itself, but rather the set of all such sums, which is usually either empty or a singleton. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- tsums = ( w e. _V , f e. _V |-> [_ ( ~P dom f i^i Fin ) / s ]_ ( ( ( TopOpen ` w ) fLimf ( s filGen ran ( z e. s |-> { y e. s | z C_ y } ) ) ) ` ( y e. s |-> ( w gsumg ( f |` y ) ) ) ) )
Theorem	tsmsfbas 21064*	The collection of all sets of the form F ( z ) = { y e. S | z C_ y }, which can be read as the set of all finite subsets of A which contain z as a subset, for each finite subset z of A, form a filter base. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- S = ( ~P A i^i Fin )   &    |- F = ( z e. S |-> { y e. S | z C_ y } )   &    |- L = ran F   &    |- ( ph -> A e. W )   =>    |- ( ph -> L e. ( fBas ` S ) )
Theorem	tsmslem1 21065	The finite partial sums of a function F are defined in a commutative monoid. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ( Base ` G )   &    |- S = ( ~P A i^i Fin )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> A e. W )   &    |- ( ph -> F : A --> B )   =>    |- ( ( ph /\ X e. S ) -> ( G gsumg ( F |` X ) ) e. B )
Theorem	tsmsval2 21066*	Definition of the topological group sum(s) of a collection F ( x ) of values in the group with index set A. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- S = ( ~P A i^i Fin )   &    |- L = ran ( z e. S |-> { y e. S | z C_ y } )   &    |- ( ph -> G e. V )   &    |- ( ph -> F e. W )   &    |- ( ph -> dom F = A )   =>    |- ( ph -> ( G tsums F ) = ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) )
Theorem	tsmsval 21067*	Definition of the topological group sum(s) of a collection F ( x ) of values in the group with index set A. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- S = ( ~P A i^i Fin )   &    |- L = ran ( z e. S |-> { y e. S | z C_ y } )   &    |- ( ph -> G e. V )   &    |- ( ph -> A e. W )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( G tsums F ) = ( ( J fLimf ( S filGen L ) ) ` ( y e. S |-> ( G gsumg ( F |` y ) ) ) ) )
Theorem	tsmspropd 21068	The group sum depends only on the base set, additive operation, and topology components. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 16504 etc. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- ( ph -> F e. V )   &    |- ( ph -> G e. W )   &    |- ( ph -> H e. X )   &    |- ( ph -> ( Base ` G ) = ( Base ` H ) )   &    |- ( ph -> ( +g ` G ) = ( +g ` H ) )   &    |- ( ph -> ( TopOpen ` G ) = ( TopOpen ` H ) )   =>    |- ( ph -> ( G tsums F ) = ( H tsums F ) )
Theorem	eltsms 21069*	The property of being a sum of the sequence F in the topological commutative monoid G. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- S = ( ~P A i^i Fin )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( C e. ( G tsums F ) <-> ( C e. B /\ A. u e. J ( C e. u -> E. z e. S A. y e. S ( z C_ y -> ( G gsumg ( F |` y ) ) e. u ) ) ) ) )
Theorem	tsmsi 21070*	The property of being a sum of the sequence F in the topological commutative monoid G. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- S = ( ~P A i^i Fin )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> C e. ( G tsums F ) )   &    |- ( ph -> U e. J )   &    |- ( ph -> C e. U )   =>    |- ( ph -> E. z e. S A. y e. S ( z C_ y -> ( G gsumg ( F |` y ) ) e. U ) )
Theorem	tsmscl 21071	A sum in a topological group is an element of the group. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( G tsums F ) C_ B )
Theorem	haustsms 21072*	In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 2-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- J = ( TopOpen ` G )   &    |- ( ph -> J e. Haus )   =>    |- ( ph -> E* x x e. ( G tsums F ) )
Theorem	haustsms2 21073	In a Hausdorff topological group, a sum has at most one limit point. (Contributed by Mario Carneiro, 13-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- J = ( TopOpen ` G )   &    |- ( ph -> J e. Haus )   =>    |- ( ph -> ( X e. ( G tsums F ) -> ( G tsums F ) = { X } ) )
Theorem	tsmscls 21074	One half of tgptsmscls 21086, true in any commutative monoid topological space. (Contributed by Mario Carneiro, 21-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> X e. ( G tsums F ) )   =>    |- ( ph -> ( ( cls ` J ) ` { X } ) C_ ( G tsums F ) )
Theorem	tsmsgsum 21075	The convergent points of a finite topological group sum are the closure of the finite group sum operation. (Contributed by Mario Carneiro, 19-Sep-2015.) (Revised by AV, 24-Jul-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> F finSupp .0. )   &    |- J = ( TopOpen ` G )   =>    |- ( ph -> ( G tsums F ) = ( ( cls ` J ) ` { ( G gsumg F ) } ) )
Theorem	tsmsid 21076	If a sum is finite, the usual sum is always a limit point of the topological sum (although it may not be the only limit point). (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by AV, 24-Jul-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> F finSupp .0. )   =>    |- ( ph -> ( G gsumg F ) e. ( G tsums F ) )
Theorem	haustsmsid 21077	In a Hausdorff topological group, a finite sum sums to exactly the usual number with no extraneous limit points. By setting the topology to the discrete topology (which is Hausdorff), this theorem can be used to turn any tsums theorem into a gsumg theorem, so that the infinite group sum operation can be viewed as a generalization of the finite group sum. (Contributed by Mario Carneiro, 2-Sep-2015.) (Revised by AV, 24-Jul-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> F finSupp .0. )   &    |- J = ( TopOpen ` G )   &    |- ( ph -> J e. Haus )   =>    |- ( ph -> ( G tsums F ) = { ( G gsumg F ) } )
Theorem	tsms0 21078*	The sum of zero is zero. (Contributed by Mario Carneiro, 18-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
|- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   =>    |- ( ph -> .0. e. ( G tsums ( x e. A |-> .0. ) ) )
Theorem	tsmssubm 21079	Evaluate an infinite group sum in a submonoid. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- ( ph -> A e. V )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> S e. (SubMnd ` G ) )   &    |- ( ph -> F : A --> S )   &    |- H = ( G ↾s S )   =>    |- ( ph -> ( H tsums F ) = ( ( G tsums F ) i^i S ) )
Theorem	tsmsres 21080	Extend an infinite group sum by padding outside with zeroes. (Contributed by Mario Carneiro, 18-Sep-2015.) (Revised by AV, 25-Jul-2019.)
|- B = ( Base ` G )   &    |- .0. = ( 0g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> ( F supp .0. ) C_ W )   =>    |- ( ph -> ( G tsums ( F |` W ) ) = ( G tsums F ) )
Theorem	tsmsf1o 21081	Re-index an infinite group sum using a bijection. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> H : C -1-1-onto-> A )   =>    |- ( ph -> ( G tsums F ) = ( G tsums ( F o. H ) ) )
Theorem	tsmsmhm 21082	Apply a continuous group homomorphism to an infinite group sum. (Contributed by Mario Carneiro, 18-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- K = ( TopOpen ` H )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopSp )   &    |- ( ph -> H e. CMnd )   &    |- ( ph -> H e. TopSp )   &    |- ( ph -> C e. ( G MndHom H ) )   &    |- ( ph -> C e. ( J Cn K ) )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> X e. ( G tsums F ) )   =>    |- ( ph -> ( C ` X ) e. ( H tsums ( C o. F ) ) )
Theorem	tsmsadd 21083	The sum of two infinite group sums. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopMnd )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> H : A --> B )   &    |- ( ph -> X e. ( G tsums F ) )   &    |- ( ph -> Y e. ( G tsums H ) )   =>    |- ( ph -> ( X .+ Y ) e. ( G tsums ( F oF .+ H ) ) )
Theorem	tsmsinv 21084	Inverse of an infinite group sum. (Contributed by Mario Carneiro, 20-Sep-2015.)
|- B = ( Base ` G )   &    |- I = ( invg ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopGrp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> X e. ( G tsums F ) )   =>    |- ( ph -> ( I ` X ) e. ( G tsums ( I o. F ) ) )
Theorem	tsmssub 21085	The difference of two infinite group sums. (Contributed by Mario Carneiro, 20-Sep-2015.)
|- B = ( Base ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopGrp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> H : A --> B )   &    |- ( ph -> X e. ( G tsums F ) )   &    |- ( ph -> Y e. ( G tsums H ) )   =>    |- ( ph -> ( X .- Y ) e. ( G tsums ( F oF .- H ) ) )
Theorem	tgptsmscls 21086	A sum in a topological group is uniquely determined up to a coset of cls ( { 0 } ), which is a normal subgroup by clsnsg 21046, 0nsg 16804. (Contributed by Mario Carneiro, 22-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopGrp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> X e. ( G tsums F ) )   =>    |- ( ph -> ( G tsums F ) = ( ( cls ` J ) ` { X } ) )
Theorem	tgptsmscld 21087	The set of limit points to an infinite sum in a topological group is closed. (Contributed by Mario Carneiro, 22-Sep-2015.)
|- B = ( Base ` G )   &    |- J = ( TopOpen ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopGrp )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   =>    |- ( ph -> ( G tsums F ) e. ( Clsd ` J ) )
Theorem	tsmssplit 21088	Split a topological group sum into two parts. (Contributed by Mario Carneiro, 19-Sep-2015.) (Proof shortened by AV, 24-Jul-2019.)
|- B = ( Base ` G )   &    |- .+ = ( +g ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopMnd )   &    |- ( ph -> A e. V )   &    |- ( ph -> F : A --> B )   &    |- ( ph -> X e. ( G tsums ( F |` C ) ) )   &    |- ( ph -> Y e. ( G tsums ( F |` D ) ) )   &    |- ( ph -> ( C i^i D ) = (/) )   &    |- ( ph -> A = ( C u. D ) )   =>    |- ( ph -> ( X .+ Y ) e. ( G tsums F ) )
Theorem	tsmsxplem1 21089*	Lemma for tsmsxp 21091. (Contributed by Mario Carneiro, 21-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopGrp )   &    |- ( ph -> A e. V )   &    |- ( ph -> C e. W )   &    |- ( ph -> F : ( A X. C ) --> B )   &    |- ( ph -> H : A --> B )   &    |- ( ( ph /\ j e. A ) -> ( H ` j ) e. ( G tsums ( k e. C |-> ( j F k ) ) ) )   &    |- J = ( TopOpen ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> L e. J )   &    |- ( ph -> .0. e. L )   &    |- ( ph -> K e. ( ~P A i^i Fin ) )   &    |- ( ph -> dom D C_ K )   &    |- ( ph -> D e. ( ~P ( A X. C ) i^i Fin ) )   =>    |- ( ph -> E. n e. ( ~P C i^i Fin ) ( ran D C_ n /\ A. x e. K ( ( H ` x ) .- ( G gsumg ( F |` ( { x } X. n ) ) ) ) e. L ) )
Theorem	tsmsxplem2 21090*	Lemma for tsmsxp 21091. (Contributed by Mario Carneiro, 21-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopGrp )   &    |- ( ph -> A e. V )   &    |- ( ph -> C e. W )   &    |- ( ph -> F : ( A X. C ) --> B )   &    |- ( ph -> H : A --> B )   &    |- ( ( ph /\ j e. A ) -> ( H ` j ) e. ( G tsums ( k e. C |-> ( j F k ) ) ) )   &    |- J = ( TopOpen ` G )   &    |- .0. = ( 0g ` G )   &    |- .+ = ( +g ` G )   &    |- .- = ( -g ` G )   &    |- ( ph -> L e. J )   &    |- ( ph -> .0. e. L )   &    |- ( ph -> K e. ( ~P A i^i Fin ) )   &    |- ( ph -> A. c e. S A. d e. T ( c .+ d ) e. U )   &    |- ( ph -> N e. ( ~P C i^i Fin ) )   &    |- ( ph -> D C_ ( K X. N ) )   &    |- ( ph -> A. x e. K ( ( H ` x ) .- ( G gsumg ( F |` ( { x } X. N ) ) ) ) e. L )   &    |- ( ph -> ( G gsumg ( F |` ( K X. N ) ) ) e. S )   &    |- ( ph -> A. g e. ( L ^m K ) ( G gsumg g ) e. T )   =>    |- ( ph -> ( G gsumg ( H |` K ) ) e. U )
Theorem	tsmsxp 21091*	Write a sum over a two-dimensional region as a double sum. This infinite group sum version of gsumxp 17534 is also known as Fubini's theorem. The converse is not necessarily true without additional assumptions. See tsmsxplem1 21089 for the main proof; this part mostly sets up the local assumptions. (Contributed by Mario Carneiro, 21-Sep-2015.)
|- B = ( Base ` G )   &    |- ( ph -> G e. CMnd )   &    |- ( ph -> G e. TopGrp )   &    |- ( ph -> A e. V )   &    |- ( ph -> C e. W )   &    |- ( ph -> F : ( A X. C ) --> B )   &    |- ( ph -> H : A --> B )   &    |- ( ( ph /\ j e. A ) -> ( H ` j ) e. ( G tsums ( k e. C |-> ( j F k ) ) ) )   =>    |- ( ph -> ( G tsums F ) C_ ( G tsums H ) )
12.2.8  Topological rings, fields, vector spaces
Syntax	ctrg 21092	The class of all topological division rings.
class TopRing
Syntax	ctdrg 21093	The class of all topological division rings.
class TopDRing
Syntax	ctlm 21094	The class of all topological modules.
class TopMod
Syntax	ctvc 21095	The class of all topological vector spaces.
class TopVec
Definition	df-trg 21096	Define a topological ring, which is a ring such that the addition is a topological group operation and the multiplication is continuous. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- TopRing = { r e. ( TopGrp i^i Ring ) | (mulGrp ` r ) e. TopMnd }
Definition	df-tdrg 21097	Define a topological division ring (which differs from a topological field only in being potentially noncommutative), which is a division ring and topological ring such that the unit group of the division ring (which is the set of nonzero elements) is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- TopDRing = { r e. ( TopRing i^i DivRing ) | ( (mulGrp ` r ) ↾s (Unit ` r ) ) e. TopGrp }
Definition	df-tlm 21098	Define a topological left module, which is just what its name suggests: instead of a group over a ring with a scalar product connecting them, it is a topological group over a topological ring with a continuous scalar product. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- TopMod = { w e. (TopMnd i^i LMod ) | ( (Scalar ` w ) e. TopRing /\ ( .sf ` w ) e. ( ( ( TopOpen ` (Scalar ` w ) ) tX ( TopOpen ` w ) ) Cn ( TopOpen ` w ) ) ) }
Definition	df-tvc 21099	Define a topological left vector space, which is a topological module over a topological division ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
|- TopVec = { w e. TopMod | (Scalar ` w ) e. TopDRing }
Theorem	istrg 21100	Express the predicate " R is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)
|- M = (mulGrp ` R )   =>    |- ( R e. TopRing <-> ( R e. TopGrp /\ R e. Ring /\ M e. TopMnd ) )