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Theorem rngmgpf 20109
Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20200 analog). (Contributed by AV, 22-Feb-2025.)
Assertion
Ref Expression
rngmgpf (mulGrp ↾ Rng):Rng⟶Smgrp

Proof of Theorem rngmgpf
StepHypRef Expression
1 fnmgp 20088 . . 3 mulGrp Fn V
2 ssv 4001 . . 3 Rng ⊆ V
3 fnssres 6679 . . 3 ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
41, 2, 3mp2an 690 . 2 (mulGrp ↾ Rng) Fn Rng
5 fvres 6915 . . . 4 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6 eqid 2725 . . . . 5 (mulGrp‘𝑎) = (mulGrp‘𝑎)
76rngmgp 20108 . . . 4 (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
85, 7eqeltrd 2825 . . 3 (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
98rgen 3052 . 2 𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10 ffnfv 7128 . 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 3050  Vcvv 3461  wss 3944  cres 5680   Fn wfn 6544  wf 6545  cfv 6549  Smgrpcsgrp 18681  mulGrpcmgp 20086  Rngcrng 20104
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 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-mgp 20087  df-rng 20105
This theorem is referenced by:  prdsrngd  20128
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