Theorem cnmpt1mulr

Description: Continuity of ring multiplication; analogue of cnmpt12f which cannot be used directly because .r is not a function. (Contributed by Mario Carneiro, 5-Oct-2015)

	Ref	Expression
Hypotheses	mulrcn.j	|- J = ( TopOpen ` R )
	cnmpt1mulr.t	|- .x. = ( .r ` R )
	cnmpt1mulr.r	|- ( ph -> R e. TopRing )
	cnmpt1mulr.k	|- ( ph -> K e. ( TopOn ` X ) )
	cnmpt1mulr.a	|- ( ph -> ( x e. X |-> A ) e. ( K Cn J ) )
	cnmpt1mulr.b	|- ( ph -> ( x e. X |-> B ) e. ( K Cn J ) )
Assertion	cnmpt1mulr	|- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( K Cn J ) )

Proof

Step	Hyp	Ref	Expression
1		mulrcn.j	|- J = ( TopOpen ` R )
2		cnmpt1mulr.t	|- .x. = ( .r ` R )
3		cnmpt1mulr.r	|- ( ph -> R e. TopRing )
4		cnmpt1mulr.k	|- ( ph -> K e. ( TopOn ` X ) )
5		cnmpt1mulr.a	|- ( ph -> ( x e. X |-> A ) e. ( K Cn J ) )
6		cnmpt1mulr.b	|- ( ph -> ( x e. X |-> B ) e. ( K Cn J ) )
7		eqid	|- ( mulGrp ` R ) = ( mulGrp ` R )
8	7 1	mgptopn	|- J = ( TopOpen ` ( mulGrp ` R ) )
9	7 2	mgpplusg	|- .x. = ( +g ` ( mulGrp ` R ) )
10	7	trgtmd	|- ( R e. TopRing -> ( mulGrp ` R ) e. TopMnd )
11	3 10	syl	|- ( ph -> ( mulGrp ` R ) e. TopMnd )
12	8 9 11 4 5 6	cnmpt1plusg	|- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( K Cn J ) )