Theorem dfur2g 13839

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 13799	. . . 4 |- mulGrp Fn _V
2		elex 2788	. . . 4 |- ( R e. V -> R e. _V )
3		funfvex 5616	. . . . 5 |- ( ( Fun mulGrp /\ R e. dom mulGrp ) -> (mulGrp ` R ) e. _V )
4	3	funfni 5395	. . . 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 2207	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
7		eqid 2207	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
8		eqid 2207	. . . 4 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
9	6, 7, 8	grpidvalg 13320	. . 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 2207	. . 3 |- (mulGrp ` R ) = (mulGrp ` R )
12		dfur2.u	. . 3 |- .1. = ( 1r ` R )
13	11, 12	ringidvalg 13838	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
14		dfur2.b	. . . . . 6 |- B = ( Base ` R )
15	11, 14	mgpbasg 13803	. . . . 5 |- ( R e. V -> B = ( Base ` (mulGrp ` R ) ) )
16	15	eleq2d 2277	. . . 4 |- ( R e. V -> ( e e. B <-> e e. ( Base ` (mulGrp ` R ) ) ) )
17		dfur2.t	. . . . . . . . 9 |- .x. = ( .r ` R )
18	11, 17	mgpplusgg 13801	. . . . . . . 8 |- ( R e. V -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	oveqd 5984	. . . . . . 7 |- ( R e. V -> ( e .x. x ) = ( e ( +g ` (mulGrp ` R ) ) x ) )
20	19	eqeq1d 2216	. . . . . 6 |- ( R e. V -> ( ( e .x. x ) = x <-> ( e ( +g ` (mulGrp ` R ) ) x ) = x ) )
21	18	oveqd 5984	. . . . . . 7 |- ( R e. V -> ( x .x. e ) = ( x ( +g ` (mulGrp ` R ) ) e ) )
22	21	eqeq1d 2216	. . . . . 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 2721	. . . 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 5273	. 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 2250	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 1373   e. wcel 2178   A.wral 2486   _Vcvv 2776   iotacio 5249   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   Basecbs 12947   +g cplusg 13024   .rcmulr 13025   0gc0g 13203  mulGrpcmgp 13797   1rcur 13836

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-addass 8062  ax-i2m1 8065  ax-0lt1 8066  ax-0id 8068  ax-rnegex 8069  ax-pre-ltirr 8072  ax-pre-ltadd 8076

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-ltxr 8147  df-inn 9072  df-2 9130  df-3 9131  df-ndx 12950  df-slot 12951  df-base 12953  df-sets 12954  df-plusg 13037  df-mulr 13038  df-0g 13205  df-mgp 13798  df-ur 13837

This theorem is referenced by: (None)