Theorem rngmgpf 13949

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 14023 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 13934	. . 3 |- mulGrp Fn _V
2		ssv 3249	. . 3 |- Rng C_ _V
3		fnssres 5445	. . 3 |- ( (mulGrp Fn _V /\ Rng C_ _V ) -> (mulGrp |` Rng ) Fn Rng )
4	1, 2, 3	mp2an 426	. 2 |- (mulGrp |` Rng ) Fn Rng
5		fvres 5663	. . . 4 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) = (mulGrp ` a ) )
6		eqid 2231	. . . . 5 |- (mulGrp ` a ) = (mulGrp ` a )
7	6	rngmgp 13948	. . . 4 |- ( a e. Rng -> (mulGrp ` a ) e. Smgrp )
8	5, 7	eqeltrd 2308	. . 3 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) e. Smgrp )
9	8	rgen 2585	. 2 |- A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp
10		ffnfv 5805	. 2 |- ( (mulGrp |` Rng ) :Rng -->Smgrp <-> ( (mulGrp |` Rng ) Fn Rng /\ A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp ) )
11	4, 9, 10	mpbir2an 950	1 |- (mulGrp |` Rng ) :Rng -->Smgrp

Syntax hints:   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   |` cres 4727   Fn wfn 5321   -->wf 5322   ` cfv 5326  Smgrpcsgrp 13483  mulGrpcmgp 13932  Rngcrng 13944

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1re 8125  ax-addrcl 8128

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-mgp 13933  df-rng 13945

This theorem is referenced by: (None)