Theorem ringideu 13205

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 2177	. . . 4 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	ringmgp 13190	. . 3 |- ( R e. Ring -> (mulGrp ` R ) e. Mnd )
3		eqid 2177	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
4		eqid 2177	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
5	3, 4	mndideu 12832	. . 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 13141	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
9		ringcl.t	. . . . . . . . 9 |- .x. = ( .r ` R )
10	1, 9	mgpplusgg 13139	. . . . . . . 8 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
11	10	oveqd 5894	. . . . . . 7 |- ( R e. Ring -> ( u .x. x ) = ( u ( +g ` (mulGrp ` R ) ) x ) )
12	11	eqeq1d 2186	. . . . . 6 |- ( R e. Ring -> ( ( u .x. x ) = x <-> ( u ( +g ` (mulGrp ` R ) ) x ) = x ) )
13	10	oveqd 5894	. . . . . . 7 |- ( R e. Ring -> ( x .x. u ) = ( x ( +g ` (mulGrp ` R ) ) u ) )
14	13	eqeq1d 2186	. . . . . 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 2685	. . . 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 2661	. . 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 2675	. . . 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 1353   e. wcel 2148   A.wral 2455   E!wreu 2457   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   .rcmulr 12539   Mndcmnd 12822  mulGrpcmgp 13135   Ringcrg 13184

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-mgp 13136  df-ring 13186

This theorem is referenced by:  isringid  13213