Metamath Proof Explorer


Theorem mulrcn

Description: The functionalization of the ring multiplication operation is a continuous function in a topological ring. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Hypotheses mulrcn.j
|- J = ( TopOpen ` R )
mulrcn.t
|- T = ( +f ` ( mulGrp ` R ) )
Assertion mulrcn
|- ( R e. TopRing -> T e. ( ( J tX J ) Cn J ) )

Proof

Step Hyp Ref Expression
1 mulrcn.j
 |-  J = ( TopOpen ` R )
2 mulrcn.t
 |-  T = ( +f ` ( mulGrp ` R ) )
3 eqid
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
4 3 trgtmd
 |-  ( R e. TopRing -> ( mulGrp ` R ) e. TopMnd )
5 3 1 mgptopn
 |-  J = ( TopOpen ` ( mulGrp ` R ) )
6 5 2 tmdcn
 |-  ( ( mulGrp ` R ) e. TopMnd -> T e. ( ( J tX J ) Cn J ) )
7 4 6 syl
 |-  ( R e. TopRing -> T e. ( ( J tX J ) Cn J ) )