Theorem rngmgpf 14176

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 14254 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 14161	. . 3 |- mulGrp Fn _V
2		ssv 3264	. . 3 |- Rng C_ _V
3		fnssres 5476	. . 3 |- ( (mulGrp Fn _V /\ Rng C_ _V ) -> (mulGrp |` Rng ) Fn Rng )
4	1, 2, 3	mp2an 426	. 2 |- (mulGrp |` Rng ) Fn Rng
5		fvres 5699	. . . 4 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) = (mulGrp ` a ) )
6		eqid 2234	. . . . 5 |- (mulGrp ` a ) = (mulGrp ` a )
7	6	rngmgp 14175	. . . 4 |- ( a e. Rng -> (mulGrp ` a ) e. Smgrp )
8	5, 7	eqeltrd 2311	. . 3 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) e. Smgrp )
9	8	rgen 2597	. 2 |- A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp
10		ffnfv 5840	. 2 |- ( (mulGrp |` Rng ) :Rng -->Smgrp <-> ( (mulGrp |` Rng ) Fn Rng /\ A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp ) )
11	4, 9, 10	mpbir2an 951	1 |- (mulGrp |` Rng ) :Rng -->Smgrp

Syntax hints:   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3214   |` cres 4756   Fn wfn 5352   -->wf 5353   ` cfv 5357  Smgrpcsgrp 13664  mulGrpcmgp 14159  Rngcrng 14171

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-mgp 14160  df-rng 14172

This theorem is referenced by: (None)