Theorem dfur2g 14266

Description: The multiplicative identity is the unique element of the ring that is left- and right-neutral on all elements under multiplication. (Contributed by Mario Carneiro, 10-Jan-2015.)

Hypotheses

Ref	Expression
dfur2.b	|- B = ( Base ` R )
dfur2.t	|- .x. = ( .r ` R )
dfur2.u	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
dfur2g	|- ( R e. V -> .1. = ( iota e ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) )

Distinct variable groups:   x, e, B   R, e, x   e, V, x

Allowed substitution hints:   .x. ( x, e)   .1. ( x, e)

Proof of Theorem dfur2g

Step	Hyp	Ref	Expression
1		fnmgp 14219	. . . 4 |- mulGrp Fn _V
2		elex 2833	. . . 4 |- ( R e. V -> R e. _V )
3		funfvex 5712	. . . . 5 |- ( ( Fun mulGrp /\ R e. dom mulGrp ) -> (mulGrp ` R ) e. _V )
4	3	funfni 5483	. . . 4 |- ( (mulGrp Fn _V /\ R e. _V ) -> (mulGrp ` R ) e. _V )
5	1, 2, 4	sylancr 418	. . 3 |- ( R e. V -> (mulGrp ` R ) e. _V )
6		eqid 2238	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
7		eqid 2238	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
8		eqid 2238	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
9	6, 7, 8	grpidvalg 13693	. . 3 |- ( (mulGrp ` R ) e. _V -> ( 0g ` (mulGrp ` R ) ) = ( iota e ( e e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( e ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) e ) = x ) ) ) )
10	5, 9	syl 14	. 2 |- ( R e. V -> ( 0g ` (mulGrp ` R ) ) = ( iota e ( e e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( e ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) e ) = x ) ) ) )
11		eqid 2238	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
12		dfur2.u	. . 3 |- .1. = ( 1r ` R )
13	11, 12	ringidvalg 14264	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
14		dfur2.b	. . . . . 6 |- B = ( Base ` R )
15	11, 14	mgpbasg 14224	. . . . 5 |- ( R e. V -> B = ( Base ` (mulGrp ` R ) ) )
16	15	eleq2d 2308	. . . 4 |- ( R e. V -> ( e e. B <-> e e. ( Base ` (mulGrp ` R ) ) ) )
17		dfur2.t	. . . . . . . . 9 |- .x. = ( .r ` R )
18	11, 17	mgpplusgg 14221	. . . . . . . 8 |- ( R e. V -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	oveqd 6102	. . . . . . 7 |- ( R e. V -> ( e .x. x ) = ( e ( +g ` (mulGrp ` R ) ) x ) )
20	19	eqeq1d 2247	. . . . . 6 |- ( R e. V -> ( ( e .x. x ) = x <-> ( e ( +g ` (mulGrp ` R ) ) x ) = x ) )
21	18	oveqd 6102	. . . . . . 7 |- ( R e. V -> ( x .x. e ) = ( x ( +g ` (mulGrp ` R ) ) e ) )
22	21	eqeq1d 2247	. . . . . 6 |- ( R e. V -> ( ( x .x. e ) = x <-> ( x ( +g ` (mulGrp ` R ) ) e ) = x ) )
23	20, 22	anbi12d 477	. . . . 5 |- ( R e. V -> ( ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> ( ( e ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) e ) = x ) ) )
24	15, 23	raleqbidv 2765	. . . 4 |- ( R e. V -> ( A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) <-> A. x e. ( Base ` (mulGrp ` R ) ) ( ( e ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) e ) = x ) ) )
25	16, 24	anbi12d 477	. . 3 |- ( R e. V -> ( ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) <-> ( e e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( e ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) e ) = x ) ) ) )
26	25	iotabidv 5360	. 2 |- ( R e. V -> ( iota e ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) = ( iota e ( e e. ( Base ` (mulGrp ` R ) ) /\ A. x e. ( Base ` (mulGrp ` R ) ) ( ( e ( +g ` (mulGrp ` R ) ) x ) = x /\ ( x ( +g ` (mulGrp ` R ) ) e ) = x ) ) ) )
27	10, 13, 26	3eqtr4d 2281	1 |- ( R e. V -> .1. = ( iota e ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   iotacio 5335   Fn wfn 5372   ` cfv 5377  (class class class)co 6085   Basecbs 13352   +g cplusg 13431   .rcmulr 13432   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-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-pre-ltirr 8291  ax-pre-ltadd 8295

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-nel 2516  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-nul 3521  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-pnf 8362  df-mnf 8363  df-ltxr 8365  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: (None)