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Metamath Proof Explorer

Theorem invrcn2

Description: The multiplicative inverse function is a continuous function from the unit group (that is, the nonzero numbers) to itself. (Contributed by Mario Carneiro, 5-Oct-2015)

Ref Expression
Hypotheses mulrcn.j 
|- J = ( TopOpen ` R )
invrcn.i 
|- I = ( invr ` R )
invrcn.u 
|- U = ( Unit ` R )
Assertion invrcn2 
|- ( R e. TopDRing -> I e. ( ( J |`t U ) Cn ( J |`t U ) ) )

Proof

Step Hyp Ref Expression
1 mulrcn.j 
 |-  J = ( TopOpen ` R )
2 invrcn.i 
 |-  I = ( invr ` R )
3 invrcn.u 
 |-  U = ( Unit ` R )
4 eqid 
 |-  ( mulGrp ` R ) = ( mulGrp ` R )
5 4 3 tdrgunit 
 |-  ( R e. TopDRing -> ( ( mulGrp ` R ) |`s U ) e. TopGrp )
6 eqid 
 |-  ( ( mulGrp ` R ) |`s U ) = ( ( mulGrp ` R ) |`s U )
7 4 1 mgptopn 
 |-  J = ( TopOpen ` ( mulGrp ` R ) )
8 6 7 resstopn 
 |-  ( J |`t U ) = ( TopOpen ` ( ( mulGrp ` R ) |`s U ) )
9 3 6 2 invrfval 
 |-  I = ( invg ` ( ( mulGrp ` R ) |`s U ) )
10 8 9 tgpinv 
 |-  ( ( ( mulGrp ` R ) |`s U ) e. TopGrp -> I e. ( ( J |`t U ) Cn ( J |`t U ) ) )
11 5 10 syl 
 |-  ( R e. TopDRing -> I e. ( ( J |`t U ) Cn ( J |`t U ) ) )