Theorem rngmgpf 13886

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 13960 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 13871	. . 3 ⊢ mulGrp Fn V
2		ssv 3246	. . 3 ⊢ Rng ⊆ V
3		fnssres 5432	. . 3 ⊢ ((mulGrp Fn V ∧ Rng ⊆ V) → (mulGrp ↾ Rng) Fn Rng)
4	1, 2, 3	mp2an 426	. 2 ⊢ (mulGrp ↾ Rng) Fn Rng
5		fvres 5647	. . . 4 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) = (mulGrp‘𝑎))
6		eqid 2229	. . . . 5 ⊢ (mulGrp‘𝑎) = (mulGrp‘𝑎)
7	6	rngmgp 13885	. . . 4 ⊢ (𝑎 ∈ Rng → (mulGrp‘𝑎) ∈ Smgrp)
8	5, 7	eqeltrd 2306	. . 3 ⊢ (𝑎 ∈ Rng → ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp)
9	8	rgen 2583	. 2 ⊢ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp
10		ffnfv 5786	. 2 ⊢ ((mulGrp ↾ Rng):Rng⟶Smgrp ↔ ((mulGrp ↾ Rng) Fn Rng ∧ ∀𝑎 ∈ Rng ((mulGrp ↾ Rng)‘𝑎) ∈ Smgrp))
11	4, 9, 10	mpbir2an 948	1 ⊢ (mulGrp ↾ Rng):Rng⟶Smgrp

Syntax hints:   ∈ wcel 2200  ∀wral 2508  Vcvv 2799   ⊆ wss 3197   ↾ cres 4718   Fn wfn 5309  ⟶wf 5310  ‘cfv 5314  Smgrpcsgrp 13420  mulGrpcmgp 13869  Rngcrng 13881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-fv 5322  df-ov 5997  df-oprab 5998  df-mpo 5999  df-inn 9099  df-2 9157  df-3 9158  df-ndx 13021  df-slot 13022  df-base 13024  df-sets 13025  df-plusg 13109  df-mulr 13110  df-mgp 13870  df-rng 13882

This theorem is referenced by: (None)