Theorem rngmgpf 20384

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20480 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 20367	. . 3 ⊢ mulGrp Fn V
2		ssv 3932	. . 3 ⊢ Rng ⊆ V
3		fnssres 6662	. . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
4	1, 2, 3	mp2an 705	. 2 ⊢ (mulGrp ↾ Rng) Fn Rng
5		fvres 6904	. . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6		eqid 2760	. . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎)
7	6	rngmgp 20383	. . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
8	5, 7	eqeltrd 2860	. . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
9	8	rgen 3078	. 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10		ffnfv 7119	. 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 2144  ∀wral 3076  Vcvv 3450   ⊆ wss 3895   ↾ cres 5652   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  Smgrpcsgrp 18912  mulGrpcmgp 20365  Rngcrng 20379

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-mgp 20366  df-rng 20380

This theorem is used by:  prdsrngd  20403