Theorem crngridl 14543

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 14087	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` O ) )
6		ssv 3249	. . . . 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 14088	. . . . 5 |- ( R e. CRing -> ( +g ` R ) = ( +g ` O ) )
10	9	oveqdr 6045	. . . 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 13214	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
13	12	slotex 13108	. . . . . 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 6051	. . . . 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 1273	. . . 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 14084	. . . . 5 |- ( ( R e. CRing /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` O ) y ) )
21	20	3expb 1230	. . . 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 14085	. . . 4 |- ( R e. CRing -> O e. _V )
24	2, 5, 7, 10, 17, 21, 22, 23	lidlrsppropdg 14508	. . 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 1397   e. wcel 2202   _Vcvv 2802   C_ wss 3200   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   CRingccrg 14009  opp_rcoppr 14079  LIdealclidl 14480  RSpancrsp 14481

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-iress 13089  df-plusg 13172  df-mulr 13173  df-sca 13175  df-vsca 13176  df-ip 13177  df-cmn 13872  df-mgp 13933  df-cring 14011  df-oppr 14080  df-lssm 14366  df-lsp 14400  df-sra 14448  df-rgmod 14449  df-lidl 14482  df-rsp 14483

This theorem is referenced by:  crng2idl  14544