Theorem dfur2g 13150

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 13137	. . . 4 |- mulGrp Fn _V
2		elex 2750	. . . 4 |- ( R e. V -> R e. _V )
3		funfvex 5534	. . . . 5 |- ( ( Fun mulGrp /\ R e. dom mulGrp ) -> (mulGrp ` R ) e. _V )
4	3	funfni 5318	. . . 4 |- ( (mulGrp Fn _V /\ R e. _V ) -> (mulGrp ` R ) e. _V )
5	1, 2, 4	sylancr 414	. . 3 |- ( R e. V -> (mulGrp ` R ) e. _V )
6		eqid 2177	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
7		eqid 2177	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
8		eqid 2177	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
9	6, 7, 8	grpidvalg 12797	. . 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 2177	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
12		dfur2.u	. . 3 |- .1. = ( 1r ` R )
13	11, 12	ringidvalg 13149	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
14		dfur2.b	. . . . . 6 |- B = ( Base ` R )
15	11, 14	mgpbasg 13141	. . . . 5 |- ( R e. V -> B = ( Base ` (mulGrp ` R ) ) )
16	15	eleq2d 2247	. . . 4 |- ( R e. V -> ( e e. B <-> e e. ( Base ` (mulGrp ` R ) ) ) )
17		dfur2.t	. . . . . . . . 9 |- .x. = ( .r ` R )
18	11, 17	mgpplusgg 13139	. . . . . . . 8 |- ( R e. V -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	oveqd 5894	. . . . . . 7 |- ( R e. V -> ( e .x. x ) = ( e ( +g ` (mulGrp ` R ) ) x ) )
20	19	eqeq1d 2186	. . . . . 6 |- ( R e. V -> ( ( e .x. x ) = x <-> ( e ( +g ` (mulGrp ` R ) ) x ) = x ) )
21	18	oveqd 5894	. . . . . . 7 |- ( R e. V -> ( x .x. e ) = ( x ( +g ` (mulGrp ` R ) ) e ) )
22	21	eqeq1d 2186	. . . . . 6 |- ( R e. V -> ( ( x .x. e ) = x <-> ( x ( +g ` (mulGrp ` R ) ) e ) = x ) )
23	20, 22	anbi12d 473	. . . . 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 2685	. . . 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 473	. . 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 5201	. 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 2220	1 |- ( R e. V -> .1. = ( iota e ( e e. B /\ A. x e. B ( ( e .x. x ) = x /\ ( x .x. e ) = x ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2739   iotacio 5178   Fn wfn 5213   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   .rcmulr 12539   0gc0g 12710  mulGrpcmgp 13135   1rcur 13147

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-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-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  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-0g 12712  df-mgp 13136  df-ur 13148

This theorem is referenced by: (None)