Theorem ringideu 14020

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 2229	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	ringmgp 14005	. . 3 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
3		eqid 2229	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
4		eqid 2229	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
5	3, 4	mndideu 13499	. . 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 13929	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
9		ringcl.t	. . . . . . . . 9 |- .x. = ( .r ` R )
10	1, 9	mgpplusgg 13927	. . . . . . . 8 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
11	10	oveqd 6030	. . . . . . 7 |- ( R e. Ring -> ( u .x. x ) = ( u ( +g ` (mulGrp ` R ) ) x ) )
12	11	eqeq1d 2238	. . . . . 6 |- ( R e. Ring -> ( ( u .x. x ) = x <-> ( u ( +g ` (mulGrp ` R ) ) x ) = x ) )
13	10	oveqd 6030	. . . . . . 7 |- ( R e. Ring -> ( x .x. u ) = ( x ( +g ` (mulGrp ` R ) ) u ) )
14	13	eqeq1d 2238	. . . . . 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 2744	. . . 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 2716	. . 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 2730	. . . 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 1395   e. wcel 2200   A.wral 2508   E!wreu 2510   ` cfv 5324  (class class class)co 6013   Basecbs 13072   +g cplusg 13150   .rcmulr 13151   Mndcmnd 13489  mulGrpcmgp 13923   Ringcrg 13999

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-pre-ltirr 8134  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-inn 9134  df-2 9192  df-3 9193  df-ndx 13075  df-slot 13076  df-base 13078  df-sets 13079  df-plusg 13163  df-mulr 13164  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-mgp 13924  df-ring 14001

This theorem is referenced by:  isringid  14028