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Theorem mgpf 15352
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 15327 . . 3  |- mulGrp  Fn  _V
2 ssv 3198 . . 3  |-  Ring  C_  _V
3 fnssres 5357 . . 3  |-  ( (mulGrp 
Fn  _V  /\  Ring  C_  _V )  ->  (mulGrp  |`  Ring )  Fn  Ring )
41, 2, 3mp2an 653 . 2  |-  (mulGrp  |`  Ring )  Fn  Ring
5 fvres 5542 . . . 4  |-  ( a  e.  Ring  ->  ( (mulGrp  |` 
Ring ) `  a
)  =  (mulGrp `  a ) )
6 eqid 2283 . . . . 5  |-  (mulGrp `  a )  =  (mulGrp `  a )
76rngmgp 15347 . . . 4  |-  ( a  e.  Ring  ->  (mulGrp `  a )  e.  Mnd )
85, 7eqeltrd 2357 . . 3  |-  ( a  e.  Ring  ->  ( (mulGrp  |` 
Ring ) `  a
)  e.  Mnd )
98rgen 2608 . 2  |-  A. a  e.  Ring  ( (mulGrp  |`  Ring ) `  a )  e.  Mnd
10 ffnfv 5685 . 2  |-  ( (mulGrp  |` 
Ring ) : Ring --> Mnd  <->  ( (mulGrp  |`  Ring )  Fn  Ring  /\ 
A. a  e.  Ring  ( (mulGrp  |`  Ring ) `  a
)  e.  Mnd )
)
114, 9, 10mpbir2an 886 1  |-  (mulGrp  |`  Ring ) : Ring --> Mnd
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   A.wral 2543   _Vcvv 2788    C_ wss 3152    |` cres 4691    Fn wfn 5250   -->wf 5251   ` cfv 5255   Mndcmnd 14361  mulGrpcmgp 15325   Ringcrg 15337
This theorem is referenced by:  prdsrngd  15395  prds1  15397
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-mgp 15326  df-rng 15340
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