Theorem mgpf 20275

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 20163	. . 3 ⊢ mulGrp Fn V
2		ssv 4033	. . 3 ⊢ Ring ⊆ V
3		fnssres 6703	. . 3 ⊢ ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
4	1, 2, 3	mp2an 691	. 2 ⊢ (mulGrp ↾ Ring) Fn Ring
5		fvres 6939	. . . 4 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6		eqid 2740	. . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎)
7	6	ringmgp 20266	. . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
8	5, 7	eqeltrd 2844	. . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
9	8	rgen 3069	. 2 ⊢ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd
10		ffnfv 7153	. 2 ⊢ ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd))
11	4, 9, 10	mpbir2an 710	1 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd

Syntax hints:   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976   ↾ cres 5702   Fn wfn 6568  ⟶wf 6569  ‘cfv 6573  Mndcmnd 18772  mulGrpcmgp 20161  Ringcrg 20260

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-mgp 20162  df-ring 20262

This theorem is referenced by:  prdsringd  20344  prds1  20346