Theorem rngmgpf 20236

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 20331 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 20219	. . 3 ⊢ mulGrp Fn V
2		ssv 3962	. . 3 ⊢ Rng ⊆ V
3		fnssres 6660	. . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
4	1, 2, 3	mp2an 704	. 2 ⊢ (mulGrp ↾ Rng) Fn Rng
5		fvres 6902	. . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6		eqid 2763	. . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎)
7	6	rngmgp 20235	. . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
8	5, 7	eqeltrd 2863	. . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
9	8	rgen 3081	. 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10		ffnfv 7116	. 2 ⊢ ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp))
11	4, 9, 10	mpbir2an 723	1 ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp

Syntax hints:   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3906   ↾ cres 5665   Fn wfn 6533  ⟶wf 6534  ‘cfv 6538  Smgrpcsgrp 18777  mulGrpcmgp 20217  Rngcrng 20231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7415  df-mgp 20218  df-rng 20232

This theorem is referenced by:  prdsrngd  20255