Theorem mgpf 19798

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 19722	. . 3 ⊢ mulGrp Fn V
2		ssv 3945	. . 3 ⊢ Ring ⊆ V
3		fnssres 6555	. . 3 ⊢ ((mulGrp Fn V ∧ Ring ⊆ V) → (mulGrp ↾ Ring) Fn Ring)
4	1, 2, 3	mp2an 689	. 2 ⊢ (mulGrp ↾ Ring) Fn Ring
5		fvres 6793	. . . 4 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) = (mulGrp‘𝑎))
6		eqid 2738	. . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎)
7	6	ringmgp 19789	. . . 4 ⊢ (𝑎 ∈ Ring → (mulGrp‘𝑎) ∈ Mnd)
8	5, 7	eqeltrd 2839	. . 3 ⊢ (𝑎 ∈ Ring → ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd)
9	8	rgen 3074	. 2 ⊢ ∀𝑎 ∈ Ring ((mulGrp ↾ Ring)‘𝑎) ∈ Mnd
10		ffnfv 6992	. 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 2106  ∀wral 3064  Vcvv 3432   ⊆ wss 3887   ↾ cres 5591   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  Mndcmnd 18385  mulGrpcmgp 19720  Ringcrg 19783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-mgp 19721  df-ring 19785

This theorem is referenced by:  prdsringd  19851  prds1  19853