Theorem ringidval 14265

Description: The value of the unity element of a ring. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
ringidval.g	|- G = (mulGrp ` R )
ringidval.u	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
ringidval	|- .1. = ( 0g ` G )

Proof of Theorem ringidval

Dummy variables x w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 5286	. . . . . . 7 |- Rel ( 0g o. mulGrp )
2		df-ur 14263	. . . . . . . 8 |- 1r = ( 0g o. mulGrp )
3	2	releqi 4858	. . . . . . 7 |- ( Rel 1r <-> Rel ( 0g o. mulGrp ) )
4	1, 3	mpbir 146	. . . . . 6 |- Rel 1r
5		relelfvdm 5727	. . . . . 6 |- ( ( Rel 1r /\ x e. ( 1r ` R ) ) -> R e. dom 1r )
6	4, 5	mpan 428	. . . . 5 |- ( x e. ( 1r ` R ) -> R e. dom 1r )
7	6	elexd 2835	. . . 4 |- ( x e. ( 1r ` R ) -> R e. _V )
8		ringidval.u	. . . 4 |- .1. = ( 1r ` R )
9	7, 8	eleq2s 2333	. . 3 |- ( x e. .1. -> R e. _V )
10		fn0g 13695	. . . . . . . . . . . . 13 |- 0g Fn _V
11		fnrel 5479	. . . . . . . . . . . . 13 |- ( 0g Fn _V -> Rel 0g )
12	10, 11	ax-mp 5	. . . . . . . . . . . 12 |- Rel 0g
13		relelfvdm 5727	. . . . . . . . . . . 12 |- ( ( Rel 0g /\ x e. ( 0g ` G ) ) -> G e. dom 0g )
14	12, 13	mpan 428	. . . . . . . . . . 11 |- ( x e. ( 0g ` G ) -> G e. dom 0g )
15	14	elexd 2835	. . . . . . . . . 10 |- ( x e. ( 0g ` G ) -> G e. _V )
16		eqid 2238	. . . . . . . . . . 11 |- ( Base ` G ) = ( Base ` G )
17		eqid 2238	. . . . . . . . . . 11 |- ( +g ` G ) = ( +g ` G )
18		eqid 2238	. . . . . . . . . . 11 |- ( 0g ` G ) = ( 0g ` G )
19	16, 17, 18	grpidvalg 13693	. . . . . . . . . 10 |- ( G e. _V -> ( 0g ` G ) = ( iota y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) ) )
20	15, 19	syl 14	. . . . . . . . 9 |- ( x e. ( 0g ` G ) -> ( 0g ` G ) = ( iota y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) ) )
21	20	eleq2d 2308	. . . . . . . 8 |- ( x e. ( 0g ` G ) -> ( x e. ( 0g ` G ) <-> x e. ( iota y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) ) ) )
22	21	ibi 176	. . . . . . 7 |- ( x e. ( 0g ` G ) -> x e. ( iota y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) ) )
23		eliotaeu 5366	. . . . . . 7 |- ( x e. ( iota y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) ) -> E! y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) )
24	22, 23	syl 14	. . . . . 6 |- ( x e. ( 0g ` G ) -> E! y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) )
25		euex 2116	. . . . . 6 |- ( E! y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) -> E. y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) )
26	24, 25	syl 14	. . . . 5 |- ( x e. ( 0g ` G ) -> E. y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) )
27		exsimpl 1670	. . . . 5 |- ( E. y ( y e. ( Base ` G ) /\ A. z e. ( Base ` G ) ( ( y ( +g ` G ) z ) = z /\ ( z ( +g ` G ) y ) = z ) ) -> E. y y e. ( Base ` G ) )
28	26, 27	syl 14	. . . 4 |- ( x e. ( 0g ` G ) -> E. y y e. ( Base ` G ) )
29	16	basm 13414	. . . . . . 7 |- ( y e. ( Base ` G ) -> E. w w e. G )
30		fnmgp 14219	. . . . . . . . . . 11 |- mulGrp Fn _V
31		fnrel 5479	. . . . . . . . . . 11 |- (mulGrp Fn _V -> Rel mulGrp )
32	30, 31	ax-mp 5	. . . . . . . . . 10 |- Rel mulGrp
33		relelfvdm 5727	. . . . . . . . . 10 |- ( ( Rel mulGrp /\ w e. (mulGrp ` R ) ) -> R e. dom mulGrp )
34	32, 33	mpan 428	. . . . . . . . 9 |- ( w e. (mulGrp ` R ) -> R e. dom mulGrp )
35		ringidval.g	. . . . . . . . 9 |- G = (mulGrp ` R )
36	34, 35	eleq2s 2333	. . . . . . . 8 |- ( w e. G -> R e. dom mulGrp )
37	36	exlimiv 1651	. . . . . . 7 |- ( E. w w e. G -> R e. dom mulGrp )
38	29, 37	syl 14	. . . . . 6 |- ( y e. ( Base ` G ) -> R e. dom mulGrp )
39	38	elexd 2835	. . . . 5 |- ( y e. ( Base ` G ) -> R e. _V )
40	39	exlimiv 1651	. . . 4 |- ( E. y y e. ( Base ` G ) -> R e. _V )
41	28, 40	syl 14	. . 3 |- ( x e. ( 0g ` G ) -> R e. _V )
42	35, 8	ringidvalg 14264	. . . 4 |- ( R e. _V -> .1. = ( 0g ` G ) )
43	42	eleq2d 2308	. . 3 |- ( R e. _V -> ( x e. .1. <-> x e. ( 0g ` G ) ) )
44	9, 41, 43	pm5.21nii 716	. 2 |- ( x e. .1. <-> x e. ( 0g ` G ) )
45	44	eqriv 2235	1 |- .1. = ( 0g ` G )

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   E.wex 1545   E!weu 2086   e. wcel 2209   A.wral 2528   _Vcvv 2821   dom cdm 4774   o. ccom 4778   Rel wrel 4779   iotacio 5335   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   0gc0g 13610  mulGrpcmgp 14217   1rcur 14262

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-inn 9305  df-2 9363  df-3 9364  df-ndx 13355  df-slot 13356  df-base 13358  df-sets 13359  df-plusg 13444  df-mulr 13445  df-0g 13612  df-mgp 14218  df-ur 14263

This theorem is used by:  assamulgscmlem1  15041