Theorem rngmgpf 14343

Description: Restricted functionality of the multiplicative group on non-unital rings (mgpf 14422 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 14326	. . 3 |- mulGrp Fn _V
2		ssv 3270	. . 3 |- Rng C_ _V
3		fnssres 5499	. . 3 |- ( (mulGrp Fn _V /\ Rng C_ _V ) -> (mulGrp |` Rng ) Fn Rng )
4	1, 2, 3	mp2an 430	. 2 |- (mulGrp |` Rng ) Fn Rng
5		fvres 5722	. . . 4 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) = (mulGrp ` a ) )
6		eqid 2238	. . . . 5 |- (mulGrp ` a ) = (mulGrp ` a )
7	6	rngmgp 14342	. . . 4 |- ( a e. Rng -> (mulGrp ` a ) e. Smgrp )
8	5, 7	eqeltrd 2315	. . 3 |- ( a e. Rng -> ( (mulGrp |` Rng ) ` a ) e. Smgrp )
9	8	rgen 2603	. 2 |- A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp
10		ffnfv 5869	. 2 |- ( (mulGrp |` Rng ) :Rng -->Smgrp <-> ( (mulGrp |` Rng ) Fn Rng /\ A. a e. Rng ( (mulGrp |` Rng ) ` a ) e. Smgrp ) )
11	4, 9, 10	mpbir2an 955	1 |- (mulGrp |` Rng ) :Rng -->Smgrp

This proof depends on syntax axioms:   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   |` cres 4778   Fn wfn 5375   -->wf 5376   ` cfv 5380  Smgrpcsgrp 13775  mulGrpcmgp 14324  Rngcrng 14338

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-plusg 13503  df-mulr 13504  df-mgp 14325  df-rng 14339

This theorem is used by: (None)