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Theorem mgpf 20245
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 20139 . . 3 mulGrp Fn V
2 ssv 4008 . . 3 Ring ⊆ V
3 fnssres 6691 . . 3 ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
41, 2, 3mp2an 692 . 2 (mulGrp ↾ Ring) Fn Ring
5 fvres 6925 . . . 4 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2737 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76ringmgp 20236 . . . 4 (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
85, 7eqeltrd 2841 . . 3 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
98rgen 3063 . 2 𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd
10 ffnfv 7139 . 2 ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd))
114, 9, 10mpbir2an 711 1 (mulGrp ↾ Ring):Ring⟶Mnd
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  wral 3061  Vcvv 3480  wss 3951  cres 5687   Fn wfn 6556  wf 6557  cfv 6561  Mndcmnd 18747  mulGrpcmgp 20137  Ringcrg 20230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-mgp 20138  df-ring 20232
This theorem is referenced by:  prdsringd  20318  prds1  20320
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