Theorem mgpf 20145

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 20033	. . 3 ⊢ mulGrp Fn V
2		ssv 3999	. . 3 ⊢ Ring ⊆ V
3		fnssres 6664	. . 3 ⊢ ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
4	1, 2, 3	mp2an 689	. 2 ⊢ (mulGrp ↾ Ring) Fn Ring
5		fvres 6901	. . . 4 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6		eqid 2724	. . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎)
7	6	ringmgp 20136	. . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
8	5, 7	eqeltrd 2825	. . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
9	8	rgen 3055	. 2 ⊢ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd
10		ffnfv 7111	. 2 ⊢ ((mulGrp ↾ Ring):Ring⟶Mnd ↔ ((mulGrp ↾ Ring) Fn Ring ∧ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd))
11	4, 9, 10	mpbir2an 708	1 ⊢ (mulGrp ↾ Ring):Ring⟶Mnd

Syntax hints:   ∈ wcel 2098  ∀wral 3053  Vcvv 3466   ⊆ wss 3941   ↾ cres 5669   Fn wfn 6529  ⟶wf 6530  ‘cfv 6534  Mndcmnd 18659  mulGrpcmgp 20031  Ringcrg 20130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-mgp 20032  df-ring 20132

This theorem is referenced by:  prdsringd  20212  prds1  20214