Theorem crngridl 14609

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 2232	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` R ) )
3		crngridl.o	. . . . 5 |- O = (oppr ` R )
4		eqid 2231	. . . . 5 |- ( Base ` R ) = ( Base ` R )
5	3, 4	opprbasg 14152	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` O ) )
6		ssv 3250	. . . . 5 |- ( Base ` R ) C_ _V
7	6	a1i 9	. . . 4 |- ( R e. CRing -> ( Base ` R ) C_ _V )
8		eqid 2231	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
9	3, 8	oppraddg 14153	. . . . 5 |- ( R e. CRing -> ( +g ` R ) = ( +g ` O ) )
10	9	oveqdr 6056	. . . 4 |- ( ( R e. CRing /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
11		simprl 531	. . . . 5 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
12		mulrslid 13278	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
13	12	slotex 13172	. . . . . 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 533	. . . . 5 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
16		ovexg 6062	. . . . 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 1274	. . . 4 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) e. _V )
18		eqid 2231	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19		eqid 2231	. . . . . 6 |- ( .r ` O ) = ( .r ` O )
20	4, 18, 3, 19	crngoppr 14149	. . . . 5 |- ( ( R e. CRing /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` O ) y ) )
21	20	3expb 1231	. . . 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 14150	. . . 4 |- ( R e. CRing -> O e. _V )
24	2, 5, 7, 10, 17, 21, 22, 23	lidlrsppropdg 14574	. . 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 2276	1 |- ( R e. CRing -> I = (LIdeal ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   C_ wss 3201   ` cfv 5333  (class class class)co 6028   Basecbs 13145   +g cplusg 13223   .rcmulr 13224   CRingccrg 14074  opp_rcoppr 14144  LIdealclidl 14546  RSpancrsp 14547

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-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-pre-ltirr 8187  ax-pre-lttrn 8189  ax-pre-ltadd 8191

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-reu 2518  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-iun 3977  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-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8258  df-mnf 8259  df-ltxr 8261  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-ndx 13148  df-slot 13149  df-base 13151  df-sets 13152  df-iress 13153  df-plusg 13236  df-mulr 13237  df-sca 13239  df-vsca 13240  df-ip 13241  df-cmn 13936  df-mgp 13998  df-cring 14076  df-oppr 14145  df-lssm 14432  df-lsp 14466  df-sra 14514  df-rgmod 14515  df-lidl 14548  df-rsp 14549

This theorem is referenced by:  crng2idl  14610