Theorem rngmgpf 20359

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20455 analog). (Contributed by AV, 22-Feb-2025.)

Assertion

Ref	Expression
rngmgpf	⊢ (mulGrp ↾ Rng):Rng⟶Smgrp

Proof of Theorem rngmgpf

Step	Hyp	Ref	Expression
1		fnmgp 20342	. . 3 ⊢ mulGrp Fn V
2		ssv 3955	. . 3 ⊢ Rng ⊆ V
3		fnssres 6654	. . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
4	1, 2, 3	mp2an 705	. 2 ⊢ (mulGrp ↾ Rng) Fn Rng
5		fvres 6896	. . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6		eqid 2761	. . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎)
7	6	rngmgp 20358	. . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
8	5, 7	eqeltrd 2861	. . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
9	8	rgen 3079	. 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10		ffnfv 7111	. 2 ⊢ ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp))
11	4, 9, 10	mpbir2an 724	1 ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp

This proof depends on syntax axioms:   ∈ wcel 2145  ∀wral 3077  Vcvv 3451   ⊆ wss 3899   ↾ cres 5653   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  Smgrpcsgrp 18887  mulGrpcmgp 20340  Rngcrng 20354

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-mgp 20341  df-rng 20355

This theorem is used by:  prdsrngd  20378