Theorem opprsubrngg 14306

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 14298	. . . 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 14298	. . . 4 |- ( x e. (SubRng ` O ) -> O e. Rng )
4		opprsubrng.o	. . . . 5 |- O = (oppr ` R )
5	4	opprrngbg 14172	. . . 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 14178	. . . . . . 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 2697	. . . . . . 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 2806	. . . . . . . . . 10 |- y e. _V
11		vex 2806	. . . . . . . . . 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 14165	. . . . . . . . . 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 2300	. . . . . . . 8 |- ( R e. Rng -> ( ( y ( .r ` O ) z ) e. x <-> ( z ( .r ` R ) y ) e. x ) )
18	17	2ralbidv 2557	. . . . . . 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 14305	. . . . 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 14171	. . . . . 6 |- ( R e. Rng -> O e. Rng )
23		eqid 2231	. . . . . . 7 |- ( Base ` O ) = ( Base ` O )
24	23, 14	issubrng2 14305	. . . . . 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 2229	1 |- ( R e. V -> (SubRng ` R ) = (SubRng ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   ` cfv 5333  (class class class)co 6028   Basecbs 13162   .rcmulr 13241  SubGrpcsubg 13834  Rngcrng 14026  opp_rcoppr 14161  SubRngcsubrng 14292

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-iress 13170  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-subg 13837  df-cmn 13953  df-abl 13954  df-mgp 14015  df-rng 14027  df-oppr 14162  df-subrng 14293

This theorem is referenced by: (None)