Metamath Proof Explorer


Theorem mgpf

Description: Restricted functionality of the multiplicative group on rings. (Contributed by Mario Carneiro, 11-Mar-2015)

Ref Expression
Assertion mgpf
|- ( mulGrp |` Ring ) : Ring --> Mnd

Proof

Step Hyp Ref Expression
1 fnmgp
 |-  mulGrp Fn _V
2 ssv
 |-  Ring C_ _V
3 fnssres
 |-  ( ( mulGrp Fn _V /\ Ring C_ _V ) -> ( mulGrp |` Ring ) Fn Ring )
4 1 2 3 mp2an
 |-  ( mulGrp |` Ring ) Fn Ring
5 fvres
 |-  ( a e. Ring -> ( ( mulGrp |` Ring ) ` a ) = ( mulGrp ` a ) )
6 eqid
 |-  ( mulGrp ` a ) = ( mulGrp ` a )
7 6 ringmgp
 |-  ( a e. Ring -> ( mulGrp ` a ) e. Mnd )
8 5 7 eqeltrd
 |-  ( a e. Ring -> ( ( mulGrp |` Ring ) ` a ) e. Mnd )
9 8 rgen
 |-  A. a e. Ring ( ( mulGrp |` Ring ) ` a ) e. Mnd
10 ffnfv
 |-  ( ( mulGrp |` Ring ) : Ring --> Mnd <-> ( ( mulGrp |` Ring ) Fn Ring /\ A. a e. Ring ( ( mulGrp |` Ring ) ` a ) e. Mnd ) )
11 4 9 10 mpbir2an
 |-  ( mulGrp |` Ring ) : Ring --> Mnd