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Theorem mgpf 18480
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 18412 . . 3 mulGrp Fn V
2 ssv 3604 . . 3 Ring ⊆ V
3 fnssres 5962 . . 3 ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
41, 2, 3mp2an 707 . 2 (mulGrp ↾ Ring) Fn Ring
5 fvres 6164 . . . 4 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2621 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76ringmgp 18474 . . . 4 (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
85, 7eqeltrd 2698 . . 3 (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
98rgen 2917 . 2 𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd
10 ffnfv 6343 . 2 ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd))
114, 9, 10mpbir2an 954 1 (mulGrp ↾ Ring):Ring⟶Mnd
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  wral 2907  Vcvv 3186  wss 3555  cres 5076   Fn wfn 5842  wf 5843  cfv 5847  Mndcmnd 17215  mulGrpcmgp 18410  Ringcrg 18468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-mgp 18411  df-ring 18470
This theorem is referenced by:  prdsringd  18533  prds1  18535
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