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Theorem mgpf 15677
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 15652 . . 3  |- mulGrp  Fn  _V
2 ssv 3370 . . 3  |-  Ring  C_  _V
3 fnssres 5560 . . 3  |-  ( (mulGrp 
Fn  _V  /\  Ring  C_  _V )  ->  (mulGrp  |`  Ring )  Fn  Ring )
41, 2, 3mp2an 655 . 2  |-  (mulGrp  |`  Ring )  Fn  Ring
5 fvres 5747 . . . 4  |-  ( a  e.  Ring  ->  ( (mulGrp  |` 
Ring ) `  a
)  =  (mulGrp `  a ) )
6 eqid 2438 . . . . 5  |-  (mulGrp `  a )  =  (mulGrp `  a )
76rngmgp 15672 . . . 4  |-  ( a  e.  Ring  ->  (mulGrp `  a )  e.  Mnd )
85, 7eqeltrd 2512 . . 3  |-  ( a  e.  Ring  ->  ( (mulGrp  |` 
Ring ) `  a
)  e.  Mnd )
98rgen 2773 . 2  |-  A. a  e.  Ring  ( (mulGrp  |`  Ring ) `  a )  e.  Mnd
10 ffnfv 5896 . 2  |-  ( (mulGrp  |` 
Ring ) : Ring --> Mnd  <->  ( (mulGrp  |`  Ring )  Fn  Ring  /\ 
A. a  e.  Ring  ( (mulGrp  |`  Ring ) `  a
)  e.  Mnd )
)
114, 9, 10mpbir2an 888 1  |-  (mulGrp  |`  Ring ) : Ring --> Mnd
Colors of variables: wff set class
Syntax hints:    e. wcel 1726   A.wral 2707   _Vcvv 2958    C_ wss 3322    |` cres 4882    Fn wfn 5451   -->wf 5452   ` cfv 5456   Mndcmnd 14686  mulGrpcmgp 15650   Ringcrg 15662
This theorem is referenced by:  prdsrngd  15720  prds1  15722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-ov 6086  df-mgp 15651  df-rng 15665
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