Theorem isringid 14037

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 2231	. . 3 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
2		eqid 2231	. . 3 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
3		eqid 2231	. . 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 14029	. . . . 5 |- ( R e. Ring -> E! y e. B A. x e. B ( ( y .x. x ) = x /\ ( x .x. y ) = x ) )
7		reurex 2752	. . . . 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 2231	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
10	9, 4	mgpbasg 13938	. . . . 5 |- ( R e. Ring -> B = ( Base ` (mulGrp ` R ) ) )
11	9, 5	mgpplusgg 13936	. . . . . . . . 9 |- ( R e. Ring -> .x. = ( +g ` (mulGrp ` R ) ) )
12	11	oveqd 6034	. . . . . . . 8 |- ( R e. Ring -> ( y .x. x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
13	12	eqeq1d 2240	. . . . . . 7 |- ( R e. Ring -> ( ( y .x. x ) = x <-> ( y ( +g ` (mulGrp ` R ) ) x ) = x ) )
14	11	oveqd 6034	. . . . . . . 8 |- ( R e. Ring -> ( x .x. y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
15	14	eqeq1d 2240	. . . . . . 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 2746	. . . . 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 2747	. . . 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 13459	. 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 2301	. . 3 |- ( R e. Ring -> ( I e. B <-> I e. ( Base ` (mulGrp ` R ) ) ) )
22	11	oveqd 6034	. . . . . 6 |- ( R e. Ring -> ( I .x. x ) = ( I ( +g ` (mulGrp ` R ) ) x ) )
23	22	eqeq1d 2240	. . . . 5 |- ( R e. Ring -> ( ( I .x. x ) = x <-> ( I ( +g ` (mulGrp ` R ) ) x ) = x ) )
24	11	oveqd 6034	. . . . . 6 |- ( R e. Ring -> ( x .x. I ) = ( x ( +g ` (mulGrp ` R ) ) I ) )
25	24	eqeq1d 2240	. . . . 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 2746	. . 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 13973	. . 3 |- ( R e. Ring -> .1. = ( 0g ` (mulGrp ` R ) ) )
31	30	eqeq1d 2240	. 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 1397   e. wcel 2202   A.wral 2510   E.wrex 2511   E!wreu 2512   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   .rcmulr 13160   0gc0g 13338  mulGrpcmgp 13932   1rcur 13971   Ringcrg 14008

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-mgp 13933  df-ur 13972  df-ring 14010

This theorem is referenced by:  imasring  14076  subrg1  14244  cnfld1  14585