Theorem opprsubrngg 13710

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 13702	. . . 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 13702	. . . 4 |- ( x e. (SubRng ` O ) -> O e. Rng )
4		opprsubrng.o	. . . . 5 |- O = (oppr ` R )
5	4	opprrngbg 13577	. . . 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 13583	. . . . . . 7 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` O ) )
8	7	eleq2d 2263	. . . . . 6 |- ( R e. Rng -> ( x e. (SubGrp ` R ) <-> x e. (SubGrp ` O ) ) )
9		ralcom 2657	. . . . . . 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 2763	. . . . . . . . . 10 |- y e. _V
11		vex 2763	. . . . . . . . . 10 |- z e. _V
12		eqid 2193	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
13		eqid 2193	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
14		eqid 2193	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
15	12, 13, 4, 14	opprmulg 13570	. . . . . . . . . 10 |- ( ( R e. Rng /\ y e. _V /\ z e. _V ) -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
16	10, 11, 15	mp3an23 1340	. . . . . . . . 9 |- ( R e. Rng -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
17	16	eleq1d 2262	. . . . . . . 8 |- ( R e. Rng -> ( ( y ( .r ` O ) z ) e. x <-> ( z ( .r ` R ) y ) e. x ) )
18	17	2ralbidv 2518	. . . . . . 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 13709	. . . . 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 13576	. . . . . 6 |- ( R e. Rng -> O e. Rng )
23		eqid 2193	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
24	23, 14	issubrng2 13709	. . . . . 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 706	. 2 |- ( R e. V -> ( x e. (SubRng ` R ) <-> x e. (SubRng ` O ) ) )
29	28	eqrdv 2191	1 |- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   _Vcvv 2760   ` cfv 5255  (class class class)co 5919   Basecbs 12621   .rcmulr 12699  SubGrpcsubg 13240  Rngcrng 13431  opp_rcoppr 13566  SubRngcsubrng 13696

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-i2m1 7979  ax-0lt1 7980  ax-0id 7982  ax-rnegex 7983  ax-pre-ltirr 7986  ax-pre-lttrn 7988  ax-pre-ltadd 7990

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-tpos 6300  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-inn 8985  df-2 9043  df-3 9044  df-ndx 12624  df-slot 12625  df-base 12627  df-sets 12628  df-iress 12629  df-plusg 12711  df-mulr 12712  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078  df-subg 13243  df-cmn 13359  df-abl 13360  df-mgp 13420  df-rng 13432  df-oppr 13567  df-subrng 13697

This theorem is referenced by: (None)