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Theorem mgpf 19303
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 19232 . . 3 mulGrp Fn V
2 ssv 3966 . . 3 Ring ⊆ V
3 fnssres 6450 . . 3 ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
41, 2, 3mp2an 691 . 2 (mulGrp ↾ Ring) Fn Ring
5 fvres 6671 . . . 4 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2822 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76ringmgp 19294 . . . 4 (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
85, 7eqeltrd 2914 . . 3 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
98rgen 3140 . 2 𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd
10 ffnfv 6864 . 2 ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd))
114, 9, 10mpbir2an 710 1 (mulGrp ↾ Ring):Ring⟶Mnd
Colors of variables: wff setvar class
Syntax hints:  wcel 2114  wral 3130  Vcvv 3469  wss 3908  cres 5534   Fn wfn 6329  wf 6330  cfv 6334  Mndcmnd 17902  mulGrpcmgp 19230  Ringcrg 19288
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-mgp 19231  df-ring 19290
This theorem is referenced by:  prdsringd  19356  prds1  19358
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