Theorem rngmgpf 13814

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 13888 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 13799	. . 3 |- mulGrp Fn _V
2		ssv 3223	. . 3 |- Rng C_ _V
3		fnssres 5408	. . 3 |- ( (mulGrp Fn _V /\ Rng C_ _V ) -> (mulGrp |` Rng ) Fn Rng )
4	1, 2, 3	mp2an 426	. 2 |- (mulGrp |` Rng ) Fn Rng
5		fvres 5623	. . . 4 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) = (mulGrp ` a ) )
6		eqid 2207	. . . . 5 |- (mulGrp ` a ) = (mulGrp ` a )
7	6	rngmgp 13813	. . . 4 |- ( a e. Rng -> (mulGrp ` a ) e. Smgrp )
8	5, 7	eqeltrd 2284	. . 3 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) e. Smgrp )
9	8	rgen 2561	. 2 |- A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp
10		ffnfv 5761	. 2 |- ( (mulGrp |` Rng ) :Rng -->Smgrp <-> ( (mulGrp |` Rng ) Fn Rng /\ A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp ) )
11	4, 9, 10	mpbir2an 945	1 |- (mulGrp |` Rng ) :Rng -->Smgrp

Syntax hints:   e. wcel 2178   A.wral 2486   _Vcvv 2776   C_ wss 3174   |` cres 4695   Fn wfn 5285   -->wf 5286   ` cfv 5290  Smgrpcsgrp 13348  mulGrpcmgp 13797  Rngcrng 13809

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1re 8054  ax-addrcl 8057

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-mgp 13798  df-rng 13810

This theorem is referenced by: (None)