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Theorem mgpf 20070
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 19988 . . 3 mulGrp Fn V
2 ssv 4006 . . 3 Ring ⊆ V
3 fnssres 6673 . . 3 ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
41, 2, 3mp2an 690 . 2 (mulGrp ↾ Ring) Fn Ring
5 fvres 6910 . . . 4 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2732 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76ringmgp 20061 . . . 4 (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
85, 7eqeltrd 2833 . . 3 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
98rgen 3063 . 2 𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd
10 ffnfv 7117 . 2 ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd))
114, 9, 10mpbir2an 709 1 (mulGrp ↾ Ring):Ring⟶Mnd
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wral 3061  Vcvv 3474  wss 3948  cres 5678   Fn wfn 6538  wf 6539  cfv 6543  Mndcmnd 18624  mulGrpcmgp 19986  Ringcrg 20055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-ov 7411  df-mgp 19987  df-ring 20057
This theorem is referenced by:  prdsringd  20133  prds1  20135
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