Theorem isringid 14169

Description: Properties showing that an element I is the unity element of a ring. (Contributed by NM, 7-Aug-2013.)

Hypotheses

Ref	Expression
rngidm.b	|- B = ( Base ` R )
rngidm.t	|- .x. = ( .r ` R )
rngidm.u	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
isringid	|- ( R e. Ring -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> .1. = I ) )

Distinct variable groups:   x, B   x, I   x, R   x, .x.   x, .1.

Proof of Theorem isringid

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2232	. . 3 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
2		eqid 2232	. . 3 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
3		eqid 2232	. . 3 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
4		rngidm.b	. . . . . 6 |- B = ( Base ` R )
5		rngidm.t	. . . . . 6 |- .x. = ( .r ` R )
6	4, 5	ringideu 14161	. . . . 5 |- ( R e. Ring -> E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
7		reurex 2763	. . . . 5 |- ( E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) -> E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
8	6, 7	syl 14	. . . 4 |- ( R e. Ring -> E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
9		eqid 2232	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
10	9, 4	mgpbasg 14070	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
11	9, 5	mgpplusgg 14068	. . . . . . . . 9 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
12	11	oveqd 6067	. . . . . . . 8 |- ( R e. Ring -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
13	12	eqeq1d 2241	. . . . . . 7 |- ( R e. Ring -> ( ( y .x. x ) = x <-> ( y ( +g ` (mulGrp ` R ) ) x ) = x ) )
14	11	oveqd 6067	. . . . . . . 8 |- ( R e. Ring -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	14	eqeq1d 2241	. . . . . . 7 |- ( R e. Ring -> ( ( x .x. y ) = x <-> ( x ( +g ` (mulGrp ` R ) ) y ) = x ) )
16	13, 15	anbi12d 473	. . . . . 6 |- ( R e. Ring -> ( ( ( y .x. x ) = x /\ ( x .x. y ) = x ) <-> ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) ) )
17	10, 16	raleqbidv 2757	. . . . 5 |- ( R e. Ring -> ( A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) ) )
18	10, 17	rexeqbidv 2758	. . . 4 |- ( R e. Ring -> ( E. y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) <-> E. y e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) ) )
19	8, 18	mpbid 147	. . 3 |- ( R e. Ring -> E. y e. ( Base ` (mulGrp ` R ) ) A. x e. ( Base ` (mulGrp ` R ) ) ( ( y ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) y ) = x ) )
20	1, 2, 3, 19	ismgmid 13590	. 2 |- ( R e. Ring -> ( ( I e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) <-> ( 0g ` (mulGrp ` R ) ) = I ) )
21	10	eleq2d 2302	. . 3 |- ( R e. Ring -> ( I e. B <-> I e. ( Base ` (mulGrp ` R ) ) ) )
22	11	oveqd 6067	. . . . . 6 |- ( R e. Ring -> ( I .x. x ) = ( I ( +g ` (mulGrp ` R ) ) x ) )
23	22	eqeq1d 2241	. . . . 5 |- ( R e. Ring -> ( ( I .x. x ) = x <-> ( I ( +g ` (mulGrp ` R ) ) x ) = x ) )
24	11	oveqd 6067	. . . . . 6 |- ( R e. Ring -> ( x .x. I ) = ( x ( +g ` (mulGrp ` R ) ) I ) )
25	24	eqeq1d 2241	. . . . 5 |- ( R e. Ring -> ( ( x .x. I ) = x <-> ( x ( +g ` (mulGrp ` R ) ) I ) = x ) )
26	23, 25	anbi12d 473	. . . 4 |- ( R e. Ring -> ( ( ( I .x. x ) = x /\ ( x .x. I ) = x ) <-> ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) )
27	10, 26	raleqbidv 2757	. . 3 |- ( R e. Ring -> ( A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) )
28	21, 27	anbi12d 473	. 2 |- ( R e. Ring -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> ( I e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( I ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) I ) = x ) ) ) )
29		rngidm.u	. . . 4 |- .1. = ( 1r ` R )
30	9, 29	ringidvalg 14105	. . 3 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
31	30	eqeq1d 2241	. 2 |- ( R e. Ring -> ( .1. = I <-> ( 0g ` (mulGrp ` R ) ) = I ) )
32	20, 28, 31	3bitr4d 220	1 |- ( R e. Ring -> ( ( I e. B /\ A. x e. B ( ( I .x. x ) = x /\ ( x .x. I ) = x ) ) <-> .1. = I ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   A.wral 2520   E.wrex 2521   E!wreu 2522   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291   0gc0g 13469  mulGrpcmgp 14064   1rcur 14103   Ringcrg 14140

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-0g 13471  df-mgm 13569  df-sgrp 13615  df-mnd 13630  df-mgp 14065  df-ur 14104  df-ring 14142

This theorem is referenced by:  imasring  14208  subrg1  14376  cnfld1  14720