Theorem crngridl 14026

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 2194	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` R ) )
3		crngridl.o	. . . . 5 |- O = (oppr ` R )
4		eqid 2193	. . . . 5 |- ( Base ` R ) = ( Base ` R )
5	3, 4	opprbasg 13571	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` O ) )
6		ssv 3201	. . . . 5 |- ( Base ` R ) C_ _V
7	6	a1i 9	. . . 4 |- ( R e. CRing -> ( Base ` R ) C_ _V )
8		eqid 2193	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
9	3, 8	oppraddg 13572	. . . . 5 |- ( R e. CRing -> ( +g ` R ) = ( +g ` O ) )
10	9	oveqdr 5946	. . . 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 12749	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
13	12	slotex 12645	. . . . . 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 5952	. . . . 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 2193	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19		eqid 2193	. . . . . 6 |- ( .r ` O ) = ( .r ` O )
20	4, 18, 3, 19	crngoppr 13568	. . . . 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 13569	. . . 4 |- ( R e. CRing -> O e. _V )
24	2, 5, 7, 10, 17, 21, 22, 23	lidlrsppropdg 13991	. . 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 2238	1 |- ( R e. CRing -> I = (LIdeal ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3153   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   CRingccrg 13493  opp_rcoppr 13563  LIdealclidl 13963  RSpancrsp 13964

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-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-addcom 7972  ax-addass 7974  ax-i2m1 7977  ax-0lt1 7978  ax-0id 7980  ax-rnegex 7981  ax-pre-ltirr 7984  ax-pre-lttrn 7986  ax-pre-ltadd 7988

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-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-tpos 6298  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-4 9043  df-5 9044  df-6 9045  df-7 9046  df-8 9047  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-iress 12626  df-plusg 12708  df-mulr 12709  df-sca 12711  df-vsca 12712  df-ip 12713  df-cmn 13356  df-mgp 13417  df-cring 13495  df-oppr 13564  df-lssm 13849  df-lsp 13883  df-sra 13931  df-rgmod 13932  df-lidl 13965  df-rsp 13966

This theorem is referenced by:  crng2idl  14027