Theorem opprsubrngg 14373

Description: Being a subring is a symmetric property. (Contributed by AV, 15-Feb-2025.)

Hypothesis

Ref	Expression
opprsubrng.o	|- O = (oppr ` R )

Assertion

Ref	Expression
opprsubrngg	|- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )

Proof of Theorem opprsubrngg

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		subrngrcl 14365	. . . 4 |- ( x e. (SubRng ` R ) -> R e. Rng )
2	1	a1i 9	. . 3 |- ( R e. V -> ( x e. (SubRng ` R ) -> R e. Rng ) )
3		subrngrcl 14365	. . . 4 |- ( x e. (SubRng ` O ) -> O e. Rng )
4		opprsubrng.o	. . . . 5 |- O = (oppr ` R )
5	4	opprrngbg 14239	. . . 4 |- ( R e. V -> ( R e. Rng <-> O e. Rng ) )
6	3, 5	imbitrrid 156	. . 3 |- ( R e. V -> ( x e. (SubRng ` O ) -> R e. Rng ) )
7	4	opprsubgg 14245	. . . . . . 7 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` O ) )
8	7	eleq2d 2304	. . . . . 6 |- ( R e. Rng -> ( x e. (SubGrp ` R ) <-> x e. (SubGrp ` O ) ) )
9		ralcom 2708	. . . . . . 7 |- ( A. z e. x A. y e. x ( z ( .r ` R ) y ) e. x <-> A. y e. x A. z e. x ( z ( .r ` R ) y ) e. x )
10		vex 2818	. . . . . . . . . 10 |- y e. _V
11		vex 2818	. . . . . . . . . 10 |- z e. _V
12		eqid 2234	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
13		eqid 2234	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
14		eqid 2234	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
15	12, 13, 4, 14	opprmulg 14232	. . . . . . . . . 10 |- ( ( R e. Rng /\ y e. _V /\ z e. _V ) -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
16	10, 11, 15	mp3an23 1366	. . . . . . . . 9 |- ( R e. Rng -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
17	16	eleq1d 2303	. . . . . . . 8 |- ( R e. Rng -> ( ( y ( .r ` O ) z ) e. x <-> ( z ( .r ` R ) y ) e. x ) )
18	17	2ralbidv 2568	. . . . . . 7 |- ( R e. Rng -> ( A. y e. x A. z e. x ( y ( .r ` O ) z ) e. x <-> A. y e. x A. z e. x ( z ( .r ` R ) y ) e. x ) )
19	9, 18	bitr4id 199	. . . . . 6 |- ( R e. Rng -> ( A. z e. x A. y e. x ( z ( .r ` R ) y ) e. x <-> A. y e. x A. z e. x ( y ( .r ` O ) z ) e. x ) )
20	8, 19	anbi12d 473	. . . . 5 |- ( R e. Rng -> ( ( x e. (SubGrp ` R ) /\ A. z e. x A. y e. x ( z ( .r ` R ) y ) e. x ) <-> ( x e. (SubGrp ` O ) /\ A. y e. x A. z e. x ( y ( .r ` O ) z ) e. x ) ) )
21	12, 13	issubrng2 14372	. . . . 5 |- ( R e. Rng -> ( x e. (SubRng ` R ) <-> ( x e. (SubGrp ` R ) /\ A. z e. x A. y e. x ( z ( .r ` R ) y ) e. x ) ) )
22	4	opprrng 14238	. . . . . 6 |- ( R e. Rng -> O e. Rng )
23		eqid 2234	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
24	23, 14	issubrng2 14372	. . . . . 6 |- ( O e. Rng -> ( x e. (SubRng ` O ) <-> ( x e. (SubGrp ` O ) /\ A. y e. x A. z e. x ( y ( .r ` O ) z ) e. x ) ) )
25	22, 24	syl 14	. . . . 5 |- ( R e. Rng -> ( x e. (SubRng ` O ) <-> ( x e. (SubGrp ` O ) /\ A. y e. x A. z e. x ( y ( .r ` O ) z ) e. x ) ) )
26	20, 21, 25	3bitr4d 220	. . . 4 |- ( R e. Rng -> ( x e. (SubRng ` R ) <-> x e. (SubRng ` O ) ) )
27	26	a1i 9	. . 3 |- ( R e. V -> ( R e. Rng -> ( x e. (SubRng ` R ) <-> x e. (SubRng ` O ) ) ) )
28	2, 6, 27	pm5.21ndd 713	. 2 |- ( R e. V -> ( x e. (SubRng ` R ) <-> x e. (SubRng ` O ) ) )
29	28	eqrdv 2232	1 |- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   ` cfv 5354  (class class class)co 6052   Basecbs 13229   .rcmulr 13308  SubGrpcsubg 13901  Rngcrng 14093  opp_rcoppr 14228  SubRngcsubrng 14359

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 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-lttrn 8243  ax-pre-ltadd 8245

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-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-tpos 6478  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-iress 13237  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-subg 13904  df-cmn 14020  df-abl 14021  df-mgp 14082  df-rng 14094  df-oppr 14229  df-subrng 14360

This theorem is referenced by: (None)