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Theorem rngmgpf 20104
Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20195 analog). (Contributed by AV, 22-Feb-2025.)
Assertion
Ref Expression
rngmgpf (mulGrp ↾ Rng):Rng⟶Smgrp

Proof of Theorem rngmgpf
StepHypRef Expression
1 fnmgp 20083 . . 3 mulGrp Fn V
2 ssv 4006 . . 3 Rng ⊆ V
3 fnssres 6683 . . 3 ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
41, 2, 3mp2an 690 . 2 (mulGrp ↾ Rng) Fn Rng
5 fvres 6921 . . . 4 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2728 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76rngmgp 20103 . . . 4 (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
85, 7eqeltrd 2829 . . 3 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
98rgen 3060 . 2 𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10 ffnfv 7134 . 2 ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp))
114, 9, 10mpbir2an 709 1 (mulGrp ↾ Rng):Rng⟶Smgrp
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  wral 3058  Vcvv 3473  wss 3949  cres 5684   Fn wfn 6548  wf 6549  cfv 6553  Smgrpcsgrp 18685  mulGrpcmgp 20081  Rngcrng 20099
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-mgp 20082  df-rng 20100
This theorem is referenced by:  prdsrngd  20123
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