MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mgpf Structured version   Unicode version

Theorem mgpf 17791
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
StepHypRef Expression
1 fnmgp 17724 . . 3  |- mulGrp  Fn  _V
2 ssv 3484 . . 3  |-  Ring  C_  _V
3 fnssres 5707 . . 3  |-  ( (mulGrp 
Fn  _V  /\  Ring  C_  _V )  ->  (mulGrp  |`  Ring )  Fn  Ring )
41, 2, 3mp2an 676 . 2  |-  (mulGrp  |`  Ring )  Fn  Ring
5 fvres 5895 . . . 4  |-  ( a  e.  Ring  ->  ( (mulGrp  |` 
Ring ) `  a
)  =  (mulGrp `  a ) )
6 eqid 2422 . . . . 5  |-  (mulGrp `  a )  =  (mulGrp `  a )
76ringmgp 17785 . . . 4  |-  ( a  e.  Ring  ->  (mulGrp `  a )  e.  Mnd )
85, 7eqeltrd 2507 . . 3  |-  ( a  e.  Ring  ->  ( (mulGrp  |` 
Ring ) `  a
)  e.  Mnd )
98rgen 2781 . 2  |-  A. a  e.  Ring  ( (mulGrp  |`  Ring ) `  a )  e.  Mnd
10 ffnfv 6064 . 2  |-  ( (mulGrp  |` 
Ring ) : Ring --> Mnd  <->  ( (mulGrp  |`  Ring )  Fn  Ring  /\ 
A. a  e.  Ring  ( (mulGrp  |`  Ring ) `  a
)  e.  Mnd )
)
114, 9, 10mpbir2an 928 1  |-  (mulGrp  |`  Ring ) : Ring --> Mnd
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1872   A.wral 2771   _Vcvv 3080    C_ wss 3436    |` cres 4855    Fn wfn 5596   -->wf 5597   ` cfv 5601   Mndcmnd 16534  mulGrpcmgp 17722   Ringcrg 17779
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-fv 5609  df-ov 6308  df-mgp 17723  df-ring 17781
This theorem is referenced by:  prdsringd  17839  prds1  17841
  Copyright terms: Public domain W3C validator