Theorem dfur2g 13518

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 13478	. . . 4 |- mulGrp Fn _V
2		elex 2774	. . . 4 |- ( R e. V -> R e. _V )
3		funfvex 5575	. . . . 5 |- ( ( Fun mulGrp /\ R e. dom mulGrp ) -> (mulGrp ` R ) e. _V )
4	3	funfni 5358	. . . 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 2196	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
7		eqid 2196	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
8		eqid 2196	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
9	6, 7, 8	grpidvalg 13016	. . 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 2196	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
12		dfur2.u	. . 3 |- .1. = ( 1r ` R )
13	11, 12	ringidvalg 13517	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
14		dfur2.b	. . . . . 6 |- B = ( Base ` R )
15	11, 14	mgpbasg 13482	. . . . 5 |- ( R e. V -> B = ( Base ` (mulGrp ` R ) ) )
16	15	eleq2d 2266	. . . 4 |- ( R e. V -> ( e e. B <-> e e. ( Base ` (mulGrp ` R ) ) ) )
17		dfur2.t	. . . . . . . . 9 |- .x. = ( .r ` R )
18	11, 17	mgpplusgg 13480	. . . . . . . 8 |- ( R e. V -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	oveqd 5939	. . . . . . 7 |- ( R e. V -> ( e .x. x ) = ( e ( +g ` (mulGrp ` R ) ) x ) )
20	19	eqeq1d 2205	. . . . . 6 |- ( R e. V -> ( ( e .x. x ) = x <-> ( e ( +g ` (mulGrp ` R ) ) x ) = x ) )
21	18	oveqd 5939	. . . . . . 7 |- ( R e. V -> ( x .x. e ) = ( x ( +g ` (mulGrp ` R ) ) e ) )
22	21	eqeq1d 2205	. . . . . 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 2709	. . . 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 5241	. 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 2239	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 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   iotacio 5217   Fn wfn 5253   ` cfv 5258  (class class class)co 5922   Basecbs 12678   +g cplusg 12755   .rcmulr 12756   0gc0g 12927  mulGrpcmgp 13476   1rcur 13515

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-pre-ltirr 7991  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-pnf 8063  df-mnf 8064  df-ltxr 8066  df-inn 8991  df-2 9049  df-3 9050  df-ndx 12681  df-slot 12682  df-base 12684  df-sets 12685  df-plusg 12768  df-mulr 12769  df-0g 12929  df-mgp 13477  df-ur 13516

This theorem is referenced by: (None)