Theorem opprsubrngg 14224

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 14216	. . . 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 14216	. . . 4 |- ( x e. (SubRng ` O ) -> O e. Rng )
4		opprsubrng.o	. . . . 5 |- O = (oppr ` R )
5	4	opprrngbg 14090	. . . 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 14096	. . . . . . 7 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` O ) )
8	7	eleq2d 2301	. . . . . 6 |- ( R e. Rng -> ( x e. (SubGrp ` R ) <-> x e. (SubGrp ` O ) ) )
9		ralcom 2696	. . . . . . 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 2805	. . . . . . . . . 10 |- y e. _V
11		vex 2805	. . . . . . . . . 10 |- z e. _V
12		eqid 2231	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
13		eqid 2231	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
14		eqid 2231	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
15	12, 13, 4, 14	opprmulg 14083	. . . . . . . . . 10 |- ( ( R e. Rng /\ y e. _V /\ z e. _V ) -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
16	10, 11, 15	mp3an23 1365	. . . . . . . . 9 |- ( R e. Rng -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
17	16	eleq1d 2300	. . . . . . . 8 |- ( R e. Rng -> ( ( y ( .r ` O ) z ) e. x <-> ( z ( .r ` R ) y ) e. x ) )
18	17	2ralbidv 2556	. . . . . . 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 14223	. . . . 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 14089	. . . . . 6 |- ( R e. Rng -> O e. Rng )
23		eqid 2231	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
24	23, 14	issubrng2 14223	. . . . . 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 712	. 2 |- ( R e. V -> ( x e. (SubRng ` R ) <-> x e. (SubRng ` O ) ) )
29	28	eqrdv 2229	1 |- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160  SubGrpcsubg 13753  Rngcrng 13944  opp_rcoppr 14079  SubRngcsubrng 14210

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-subg 13756  df-cmn 13872  df-abl 13873  df-mgp 13933  df-rng 13945  df-oppr 14080  df-subrng 14211

This theorem is referenced by: (None)