Theorem ringideu 13513

Description: The unity element of a ring is unique. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ringcl.b	|- B = ( Base ` R )
ringcl.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
ringideu	|- ( R e. Ring -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )

Distinct variable groups:   x, B   x, R, u   u, B   u, R   u, .x. , x

Proof of Theorem ringideu

Step	Hyp	Ref	Expression
1		eqid 2193	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	ringmgp 13498	. . 3 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
3		eqid 2193	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
4		eqid 2193	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
5	3, 4	mndideu 13007	. . 3 |- ( (mulGrp ` R ) e. Mnd -> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) )
6	2, 5	syl 14	. 2 |- ( R e. Ring -> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) )
7		ringcl.b	. . . . . 6 |- B = ( Base ` R )
8	1, 7	mgpbasg 13422	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
9		ringcl.t	. . . . . . . . 9 |- .x. = ( .r ` R )
10	1, 9	mgpplusgg 13420	. . . . . . . 8 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
11	10	oveqd 5935	. . . . . . 7 |- ( R e. Ring -> ( u .x. x ) = ( u ( +g ` (mulGrp ` R ) ) x ) )
12	11	eqeq1d 2202	. . . . . 6 |- ( R e. Ring -> ( ( u .x. x ) = x <-> ( u ( +g ` (mulGrp ` R ) ) x ) = x ) )
13	10	oveqd 5935	. . . . . . 7 |- ( R e. Ring -> ( x .x. u ) = ( x ( +g ` (mulGrp ` R ) ) u ) )
14	13	eqeq1d 2202	. . . . . 6 |- ( R e. Ring -> ( ( x .x. u ) = x <-> ( x ( +g ` (mulGrp ` R ) ) u ) = x ) )
15	12, 14	anbi12d 473	. . . . 5 |- ( R e. Ring -> ( ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
16	8, 15	raleqbidv 2706	. . . 4 |- ( R e. Ring -> ( A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
17	16	reubidv 2678	. . 3 |- ( R e. Ring -> ( E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> E! u e. B A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
18		reueq1 2692	. . . 4 |- ( B = ( Base ` (mulGrp ` R ) ) -> ( E! u e. B A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) <-> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
19	8, 18	syl 14	. . 3 |- ( R e. Ring -> ( E! u e. B A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) <-> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
20	17, 19	bitrd 188	. 2 |- ( R e. Ring -> ( E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) <-> E! u e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( u ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) u ) = x ) ) )
21	6, 20	mpbird 167	1 |- ( R e. Ring -> E! u e. B A. x e. B ( ( u .x. x ) = x /\ ( x .x. u ) = x ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   A.wral 2472   E!wreu 2474   ` cfv 5254  (class class class)co 5918   Basecbs 12618   +g cplusg 12695   .rcmulr 12696   Mndcmnd 12997  mulGrpcmgp 13416   Ringcrg 13492

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-sep 4147  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-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-rmo 2480  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-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-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-pnf 8056  df-mnf 8057  df-ltxr 8059  df-inn 8983  df-2 9041  df-3 9042  df-ndx 12621  df-slot 12622  df-base 12624  df-sets 12625  df-plusg 12708  df-mulr 12709  df-mgm 12939  df-sgrp 12985  df-mnd 12998  df-mgp 13417  df-ring 13494

This theorem is referenced by:  isringid  13521