Theorem opprsubrngg 14088

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 14080	. . . 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 14080	. . . 4 |- ( x e. (SubRng ` O ) -> O e. Rng )
4		opprsubrng.o	. . . . 5 |- O = (oppr ` R )
5	4	opprrngbg 13955	. . . 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 13961	. . . . . . 7 |- ( R e. Rng -> (SubGrp ` R ) = (SubGrp ` O ) )
8	7	eleq2d 2277	. . . . . 6 |- ( R e. Rng -> ( x e. (SubGrp ` R ) <-> x e. (SubGrp ` O ) ) )
9		ralcom 2671	. . . . . . 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 2779	. . . . . . . . . 10 |- y e. _V
11		vex 2779	. . . . . . . . . 10 |- z e. _V
12		eqid 2207	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
13		eqid 2207	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
14		eqid 2207	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
15	12, 13, 4, 14	opprmulg 13948	. . . . . . . . . 10 |- ( ( R e. Rng /\ y e. _V /\ z e. _V ) -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
16	10, 11, 15	mp3an23 1342	. . . . . . . . 9 |- ( R e. Rng -> ( y ( .r ` O ) z ) = ( z ( .r ` R ) y ) )
17	16	eleq1d 2276	. . . . . . . 8 |- ( R e. Rng -> ( ( y ( .r ` O ) z ) e. x <-> ( z ( .r ` R ) y ) e. x ) )
18	17	2ralbidv 2532	. . . . . . 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 14087	. . . . 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 13954	. . . . . 6 |- ( R e. Rng -> O e. Rng )
23		eqid 2207	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
24	23, 14	issubrng2 14087	. . . . . 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 707	. 2 |- ( R e. V -> ( x e. (SubRng ` R ) <-> x e. (SubRng ` O ) ) )
29	28	eqrdv 2205	1 |- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   A.wral 2486   _Vcvv 2776   ` cfv 5290  (class class class)co 5967   Basecbs 12947   .rcmulr 13025  SubGrpcsubg 13618  Rngcrng 13809  opp_rcoppr 13944  SubRngcsubrng 14074

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-lttrn 8074  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-tpos 6354  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-iress 12955  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgm 13303  df-sgrp 13349  df-mnd 13364  df-grp 13450  df-subg 13621  df-cmn 13737  df-abl 13738  df-mgp 13798  df-rng 13810  df-oppr 13945  df-subrng 14075

This theorem is referenced by: (None)