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

Theorem rngmgpf 20059
Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20150 analog). (Contributed by AV, 22-Feb-2025.)
Assertion
Ref Expression
rngmgpf (mulGrp ↾ Rng):Rng⟶Smgrp

Proof of Theorem rngmgpf
StepHypRef Expression
1 fnmgp 20038 . . 3 mulGrp Fn V
2 ssv 4001 . . 3 Rng ⊆ V
3 fnssres 6666 . . 3 ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
41, 2, 3mp2an 689 . 2 (mulGrp ↾ Rng) Fn Rng
5 fvres 6903 . . . 4 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2726 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76rngmgp 20058 . . . 4 (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
85, 7eqeltrd 2827 . . 3 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
98rgen 3057 . 2 𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10 ffnfv 7113 . 2 ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp))
114, 9, 10mpbir2an 708 1 (mulGrp ↾ Rng):Rng⟶Smgrp
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  wral 3055  Vcvv 3468  wss 3943  cres 5671   Fn wfn 6531  wf 6532  cfv 6536  Smgrpcsgrp 18648  mulGrpcmgp 20036  Rngcrng 20054
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 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
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 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-mgp 20037  df-rng 20055
This theorem is referenced by:  prdsrngd  20078
  Copyright terms: Public domain W3C validator