MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rngmgpf Structured version   Visualization version   GIF version

Theorem rngmgpf 20155
Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20246 analog). (Contributed by AV, 22-Feb-2025.)
Assertion
Ref Expression
rngmgpf (mulGrp ↾ Rng):Rng⟶Smgrp

Proof of Theorem rngmgpf
StepHypRef Expression
1 fnmgp 20140 . . 3 mulGrp Fn V
2 ssv 4007 . . 3 Rng ⊆ V
3 fnssres 6690 . . 3 ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
41, 2, 3mp2an 692 . 2 (mulGrp ↾ Rng) Fn Rng
5 fvres 6924 . . . 4 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2736 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76rngmgp 20154 . . . 4 (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
85, 7eqeltrd 2840 . . 3 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
98rgen 3062 . 2 𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10 ffnfv 7138 . 2 ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp))
114, 9, 10mpbir2an 711 1 (mulGrp ↾ Rng):Rng⟶Smgrp
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  wral 3060  Vcvv 3479  wss 3950  cres 5686   Fn wfn 6555  wf 6556  cfv 6560  Smgrpcsgrp 18732  mulGrpcmgp 20138  Rngcrng 20150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-mgp 20139  df-rng 20151
This theorem is referenced by:  prdsrngd  20174
  Copyright terms: Public domain W3C validator