Theorem dfur2g 14123

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 14083	. . . 4 |- mulGrp Fn _V
2		elex 2827	. . . 4 |- ( R e. V -> R e. _V )
3		funfvex 5689	. . . . 5 |- ( ( Fun mulGrp /\ R e. dom mulGrp ) -> (mulGrp ` R ) e. _V )
4	3	funfni 5460	. . . 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 2234	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
7		eqid 2234	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
8		eqid 2234	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
9	6, 7, 8	grpidvalg 13603	. . 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 2234	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
12		dfur2.u	. . 3 |- .1. = ( 1r ` R )
13	11, 12	ringidvalg 14122	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
14		dfur2.b	. . . . . 6 |- B = ( Base ` R )
15	11, 14	mgpbasg 14087	. . . . 5 |- ( R e. V -> B = ( Base ` (mulGrp ` R ) ) )
16	15	eleq2d 2304	. . . 4 |- ( R e. V -> ( e e. B <-> e e. ( Base ` (mulGrp ` R ) ) ) )
17		dfur2.t	. . . . . . . . 9 |- .x. = ( .r ` R )
18	11, 17	mgpplusgg 14085	. . . . . . . 8 |- ( R e. V -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	oveqd 6069	. . . . . . 7 |- ( R e. V -> ( e .x. x ) = ( e ( +g ` (mulGrp ` R ) ) x ) )
20	19	eqeq1d 2243	. . . . . 6 |- ( R e. V -> ( ( e .x. x ) = x <-> ( e ( +g ` (mulGrp ` R ) ) x ) = x ) )
21	18	oveqd 6069	. . . . . . 7 |- ( R e. V -> ( x .x. e ) = ( x ( +g ` (mulGrp ` R ) ) e ) )
22	21	eqeq1d 2243	. . . . . 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 2759	. . . 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 5337	. 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 2277	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 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   iotacio 5312   Fn wfn 5349   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   .rcmulr 13308   0gc0g 13486  mulGrpcmgp 14081   1rcur 14120

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgp 14082  df-ur 14121

This theorem is referenced by: (None)