Theorem mgpf 18559

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

Assertion

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

Proof of Theorem mgpf

Step	Hyp	Ref	Expression
1		fnmgp 18491	. . 3 |- mulGrp Fn _V
2		ssv 3625	. . 3 |- Ring C_ _V
3		fnssres 6004	. . 3 |- ( (mulGrp Fn _V /\ Ring C_ _V ) -> (mulGrp |` Ring ) Fn Ring )
4	1, 2, 3	mp2an 708	. 2 |- (mulGrp |` Ring ) Fn Ring
5		fvres 6207	. . . 4 |- ( a e. Ring -> ( (mulGrp |` Ring ) ` a ) = (mulGrp ` a ) )
6		eqid 2622	. . . . 5 |- (mulGrp ` a ) = (mulGrp ` a )
7	6	ringmgp 18553	. . . 4 |- ( a e. Ring -> (mulGrp ` a ) e. Mnd )
8	5, 7	eqeltrd 2701	. . 3 |- ( a e. Ring -> ( (mulGrp |` Ring ) ` a ) e. Mnd )
9	8	rgen 2922	. 2 |- A. a e. Ring ( (mulGrp |` Ring ) ` a ) e. Mnd
10		ffnfv 6388	. 2 |- ( (mulGrp |` Ring ) : Ring --> Mnd <-> ( (mulGrp |` Ring ) Fn Ring /\ A. a e. Ring ( (mulGrp |` Ring ) ` a ) e. Mnd ) )
11	4, 9, 10	mpbir2an 955	1 |- (mulGrp |` Ring ) : Ring --> Mnd

Syntax hints:   e. wcel 1990   A.wral 2912   _Vcvv 3200   C_ wss 3574   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888   Mndcmnd 17294  mulGrpcmgp 18489   Ringcrg 18547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-mgp 18490  df-ring 18549

This theorem is referenced by:  prdsringd  18612  prds1  18614