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

Step	Hyp	Ref	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)
4	1, 2, 3	mp2an 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)
7	6	ringmgp 18474	. . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
8	5, 7	eqeltrd 2698	. . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
9	8	rgen 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))
11	4, 9, 10	mpbir2an 954	1 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd

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