Theorem crngridl 14976

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 2239	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` R ) )
3		crngridl.o	. . . . 5 |- O = (oppr ` R )
4		eqid 2238	. . . . 5 |- ( Base ` R ) = ( Base ` R )
5	3, 4	opprbasg 14489	. . . 4 |- ( R e. CRing -> ( Base ` R ) = ( Base ` O ) )
6		ssv 3270	. . . . 5 |- ( Base ` R ) C_ _V
7	6	a1i 9	. . . 4 |- ( R e. CRing -> ( Base ` R ) C_ _V )
8		eqid 2238	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
9	3, 8	oppraddg 14490	. . . . 5 |- ( R e. CRing -> ( +g ` R ) = ( +g ` O ) )
10	9	oveqdr 6116	. . . 4 |- ( ( R e. CRing /\ ( x e. _V /\ y e. _V ) ) -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
11		simprl 535	. . . . 5 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> x e. ( Base ` R ) )
12		mulrslid 13545	. . . . . . 7 |- ( .r = Slot ( .r ` ndx ) /\ ( .r ` ndx ) e. NN )
13	12	slotex 13437	. . . . . 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 537	. . . . 5 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> y e. ( Base ` R ) )
16		ovexg 6122	. . . . 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 1278	. . . 4 |- ( ( R e. CRing /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) ) ) -> ( x ( .r ` R ) y ) e. _V )
18		eqid 2238	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
19		eqid 2238	. . . . . 6 |- ( .r ` O ) = ( .r ` O )
20	4, 18, 3, 19	crngoppr 14486	. . . . 5 |- ( ( R e. CRing /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` R ) y ) = ( x ( .r ` O ) y ) )
21	20	3expb 1235	. . . 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 14487	. . . 4 |- ( R e. CRing -> O e. _V )
24	2, 5, 7, 10, 17, 21, 22, 23	lidlrsppropdg 14941	. . 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 2283	1 |- ( R e. CRing -> I = (LIdeal ` O ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   C_ wss 3220   ` cfv 5380  (class class class)co 6088   Basecbs 13410   +g cplusg 13490   .rcmulr 13491   CRingccrg 14408  opp_rcoppr 14481  LIdealclidl 14913  RSpancrsp 14914

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-lttrn 8299  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-tpos 6520  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-iress 13418  df-plusg 13503  df-mulr 13504  df-sca 13506  df-vsca 13507  df-ip 13508  df-cmn 14179  df-mgp 14325  df-cring 14410  df-oppr 14482  df-lssm 14799  df-lsp 14833  df-sra 14881  df-rgmod 14882  df-lidl 14915  df-rsp 14916

This theorem is used by:  crng2idl  14977