Theorem tgpmulg2

Description: 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 to write the left topology as a subset of the complex numbers. (Contributed by Mario Carneiro, 19-Sep-2015)

	Ref	Expression
Hypotheses	tgpmulg.j	|- J = ( TopOpen ` G )
	tgpmulg.t	|- .x. = ( .g ` G )
Assertion	tgpmulg2	|- ( G e. TopGrp -> .x. e. ( ( ~P ZZ tX J ) Cn J ) )

Proof

Step	Hyp	Ref	Expression
1		tgpmulg.j	|- J = ( TopOpen ` G )
2		tgpmulg.t	|- .x. = ( .g ` G )
3		zex	|- ZZ e. _V
4	3	a1i	|- ( G e. TopGrp -> ZZ e. _V )
5		eqid	|- ( Base ` G ) = ( Base ` G )
6	1 5	tgptopon	|- ( G e. TopGrp -> J e. ( TopOn ` ( Base ` G ) ) )
7		topontop	|- ( J e. ( TopOn ` ( Base ` G ) ) -> J e. Top )
8	6 7	syl	|- ( G e. TopGrp -> J e. Top )
9	5 2	mulgfn	|- .x. Fn ( ZZ X. ( Base ` G ) )
10	9	a1i	|- ( G e. TopGrp -> .x. Fn ( ZZ X. ( Base ` G ) ) )
11	1 2 5	tgpmulg	|- ( ( G e. TopGrp /\ n e. ZZ ) -> ( x e. ( Base ` G ) |-> ( n .x. x ) ) e. ( J Cn J ) )
12	4 6 8 10 11	txdis1cn	|- ( G e. TopGrp -> .x. e. ( ( ~P ZZ tX J ) Cn J ) )