Theorem crngridl 13869

Description: In a commutative ring, the left and right ideals coincide. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
crng2idl.i	|- I = (LIdeal ` R )
crngridl.o	|- O = (oppr ` R )

Assertion

Ref	Expression
crngridl	|- ( R e. CRing -> I = (LIdeal ` O ) )

Proof of Theorem crngridl

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

Step	Hyp	Ref	Expression
1		crng2idl.i	. 2 |- I = (LIdeal ` R )
2		eqidd 2190	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` R ) )
3		crngridl.o	. . . . 5 |- O = (oppr ` R )
4		eqid 2189	. . . . 5 |- ( Base ` R ) = ( Base ` R )
5	3, 4	opprbasg 13450	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` O ) )
6		ssv 3192	. . . . 5 |- ( Base ` R ) C_ _V
7	6	a1i 9	. . . 4 |- ( R e. CRing -> ( Base ` R ) C_ _V )
8		eqid 2189	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
9	3, 8	oppraddg 13451	. . . . 5 |- ( R e. CRing -> ( +g ` R ) = ( +g ` O ) )
10	9	oveqdr 5928	. . . 4 |- ( ( R e. CRing /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
11		simprl 529	. . . . 5 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
12		mulrslid 12654	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
13	12	slotex 12550	. . . . . 6 |- ( R e. CRing -> ( .r ` R ) e. _V )
14	13	adantr 276	. . . . 5 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( .r ` R ) e. _V )
15		simprr 531	. . . . 5 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
16		ovexg 5934	. . . . 5 |- ( ( x e. ( Base ` R ) /\ ( .r ` R ) e. _V /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) e. _V )
17	11, 14, 15, 16	syl3anc 1249	. . . 4 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) e. _V )
18		eqid 2189	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19		eqid 2189	. . . . . 6 |- ( .r ` O ) = ( .r ` O )
20	4, 18, 3, 19	crngoppr 13447	. . . . 5 |- ( ( R e. CRing /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` O ) y ) )
21	20	3expb 1206	. . . 4 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` O ) y ) )
22		id 19	. . . 4 |- ( R e. CRing -> R e. CRing )
23	3	opprex 13448	. . . 4 |- ( R e. CRing -> O e. _V )
24	2, 5, 7, 10, 17, 21, 22, 23	lidlrsppropdg 13836	. . 3 |- ( R e. CRing -> ( (LIdeal ` R ) = (LIdeal ` O ) /\ (RSpan ` R ) = (RSpan ` O ) ) )
25	24	simpld 112	. 2 |- ( R e. CRing -> (LIdeal ` R ) = (LIdeal ` O ) )
26	1, 25	eqtrid 2234	1 |- ( R e. CRing -> I = (LIdeal ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   _Vcvv 2752   C_ wss 3144   ` cfv 5238  (class class class)co 5900   Basecbs 12523   +g cplusg 12600   .rcmulr 12601   CRingccrg 13376  opp_rcoppr 13442  LIdealclidl 13808  RSpancrsp 13809

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-addcom 7946  ax-addass 7948  ax-i2m1 7951  ax-0lt1 7952  ax-0id 7954  ax-rnegex 7955  ax-pre-ltirr 7958  ax-pre-lttrn 7960  ax-pre-ltadd 7962

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-tpos 6274  df-pnf 8029  df-mnf 8030  df-ltxr 8032  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-5 9016  df-6 9017  df-7 9018  df-8 9019  df-ndx 12526  df-slot 12527  df-base 12529  df-sets 12530  df-iress 12531  df-plusg 12613  df-mulr 12614  df-sca 12616  df-vsca 12617  df-ip 12618  df-cmn 13250  df-mgp 13300  df-cring 13378  df-oppr 13443  df-lssm 13694  df-lsp 13728  df-sra 13776  df-rgmod 13777  df-lidl 13810  df-rsp 13811

This theorem is referenced by:  crng2idl  13870