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Theorem rngmgpf 20175
Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20266 analog). (Contributed by AV, 22-Feb-2025.)
Assertion
Ref Expression
rngmgpf (mulGrp ↾ Rng):Rng⟶Smgrp

Proof of Theorem rngmgpf
StepHypRef Expression
1 fnmgp 20154 . . 3 mulGrp Fn V
2 ssv 4020 . . 3 Rng ⊆ V
3 fnssres 6692 . . 3 ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
41, 2, 3mp2an 692 . 2 (mulGrp ↾ Rng) Fn Rng
5 fvres 6926 . . . 4 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2735 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76rngmgp 20174 . . . 4 (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
85, 7eqeltrd 2839 . . 3 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
98rgen 3061 . 2 𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10 ffnfv 7139 . 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 2106  wral 3059  Vcvv 3478  wss 3963  cres 5691   Fn wfn 6558  wf 6559  cfv 6563  Smgrpcsgrp 18744  mulGrpcmgp 20152  Rngcrng 20170
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-mgp 20153  df-rng 20171
This theorem is referenced by:  prdsrngd  20194
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