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Theorem mgpf 20266
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 20154 . . 3 mulGrp Fn V
2 ssv 4020 . . 3 Ring ⊆ V
3 fnssres 6692 . . 3 ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
41, 2, 3mp2an 692 . 2 (mulGrp ↾ Ring) Fn Ring
5 fvres 6926 . . . 4 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2735 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76ringmgp 20257 . . . 4 (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
85, 7eqeltrd 2839 . . 3 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
98rgen 3061 . 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 2106  wral 3059  Vcvv 3478  wss 3963  cres 5691   Fn wfn 6558  wf 6559  cfv 6563  Mndcmnd 18760  mulGrpcmgp 20152  Ringcrg 20251
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-mgp 20153  df-ring 20253
This theorem is referenced by:  prdsringd  20335  prds1  20337
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