Theorem rngmgpf 13288

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 13362 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 13273	. . 3 |- mulGrp Fn _V
2		ssv 3192	. . 3 |- Rng C_ _V
3		fnssres 5348	. . 3 |- ( (mulGrp Fn _V /\ Rng C_ _V ) -> (mulGrp |` Rng ) Fn Rng )
4	1, 2, 3	mp2an 426	. 2 |- (mulGrp |` Rng ) Fn Rng
5		fvres 5558	. . . 4 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) = (mulGrp ` a ) )
6		eqid 2189	. . . . 5 |- (mulGrp ` a ) = (mulGrp ` a )
7	6	rngmgp 13287	. . . 4 |- ( a e. Rng -> (mulGrp ` a ) e. Smgrp )
8	5, 7	eqeltrd 2266	. . 3 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) e. Smgrp )
9	8	rgen 2543	. 2 |- A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp
10		ffnfv 5694	. 2 |- ( (mulGrp |` Rng ) :Rng -->Smgrp <-> ( (mulGrp |` Rng ) Fn Rng /\ A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp ) )
11	4, 9, 10	mpbir2an 944	1 |- (mulGrp |` Rng ) :Rng -->Smgrp

Syntax hints:   e. wcel 2160   A.wral 2468   _Vcvv 2752   C_ wss 3144   |` cres 4646   Fn wfn 5230   -->wf 5231   ` cfv 5235  Smgrpcsgrp 12861  mulGrpcmgp 13271  Rngcrng 13283

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-1re 7934  ax-addrcl 7937

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5898  df-oprab 5899  df-mpo 5900  df-inn 8949  df-2 9007  df-3 9008  df-ndx 12514  df-slot 12515  df-base 12517  df-sets 12518  df-plusg 12599  df-mulr 12600  df-mgp 13272  df-rng 13284

This theorem is referenced by: (None)