Theorem oppr1g 13449

Description: Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)

Hypotheses

Ref	Expression
opprbas.1	|- O = (oppr ` R )
oppr1.2	|- .1. = ( 1r ` R )

Assertion

Ref	Expression
oppr1g	|- ( R e. V -> .1. = ( 1r ` O ) )

Proof of Theorem oppr1g

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2189	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
2		eqid 2189	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
3		opprbas.1	. . . . . . . . . . 11 |- O = (oppr ` R )
4		eqid 2189	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
5	1, 2, 3, 4	opprmulg 13438	. . . . . . . . . 10 |- ( ( R e. V /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
6	5	3expa 1205	. . . . . . . . 9 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
7	6	eqeq1d 2198	. . . . . . . 8 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( x ( .r ` O ) y ) = y <-> ( y ( .r ` R ) x ) = y ) )
8		simpll 527	. . . . . . . . . 10 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> R e. V )
9		simpr 110	. . . . . . . . . 10 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> y e. ( Base ` R ) )
10		simplr 528	. . . . . . . . . 10 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
11	1, 2, 3, 4	opprmulg 13438	. . . . . . . . . 10 |- ( ( R e. V /\ y e. ( Base ` R ) /\ x e. ( Base ` R ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
12	8, 9, 10, 11	syl3anc 1249	. . . . . . . . 9 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
13	12	eqeq1d 2198	. . . . . . . 8 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( y ( .r ` O ) x ) = y <-> ( x ( .r ` R ) y ) = y ) )
14	7, 13	anbi12d 473	. . . . . . 7 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) <-> ( ( y ( .r ` R ) x ) = y /\ ( x ( .r ` R ) y ) = y ) ) )
15	14	biancomd 271	. . . . . 6 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) <-> ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
16	15	ralbidva 2486	. . . . 5 |- ( ( R e. V /\ x e. ( Base ` R ) ) -> ( A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) <-> A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
17	16	riotabidva 5869	. . . 4 |- ( R e. V -> ( iota_ x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) = ( iota_ x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
18		df-riota 5852	. . . 4 |- ( iota_ x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) )
19		df-riota 5852	. . . 4 |- ( iota_ x e. ( Base ` R ) A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) )
20	17, 18, 19	3eqtr3g 2245	. . 3 |- ( R e. V -> ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) ) )
21	3	opprex 13440	. . . . 5 |- ( R e. V -> O e. _V )
22		eqid 2189	. . . . . 6 |- (mulGrp ` O ) = (mulGrp ` O )
23	22	mgpex 13296	. . . . 5 |- ( O e. _V -> (mulGrp ` O ) e. _V )
24		eqid 2189	. . . . . 6 |- ( Base ` (mulGrp ` O ) ) = ( Base ` (mulGrp ` O ) )
25		eqid 2189	. . . . . 6 |- ( +g ` (mulGrp ` O ) ) = ( +g ` (mulGrp ` O ) )
26		eqid 2189	. . . . . 6 |- ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` O ) )
27	24, 25, 26	grpidvalg 12852	. . . . 5 |- ( (mulGrp ` O ) e. _V -> ( 0g ` (mulGrp ` O ) ) = ( iota x ( x e. ( Base ` (mulGrp ` O ) ) /\ A. y e. ( Base ` (mulGrp ` O ) ) ( ( x ( +g ` (mulGrp ` O ) ) y ) = y /\ ( y ( +g ` (mulGrp ` O ) ) x ) = y ) ) ) )
28	21, 23, 27	3syl 17	. . . 4 |- ( R e. V -> ( 0g ` (mulGrp ` O ) ) = ( iota x ( x e. ( Base ` (mulGrp ` O ) ) /\ A. y e. ( Base ` (mulGrp ` O ) ) ( ( x ( +g ` (mulGrp ` O ) ) y ) = y /\ ( y ( +g ` (mulGrp ` O ) ) x ) = y ) ) ) )
29	3, 1	opprbasg 13442	. . . . . . . 8 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
30		eqid 2189	. . . . . . . . . 10 |- ( Base ` O ) = ( Base ` O )
31	22, 30	mgpbasg 13297	. . . . . . . . 9 |- ( O e. _V -> ( Base ` O ) = ( Base ` (mulGrp ` O ) ) )
32	21, 31	syl 14	. . . . . . . 8 |- ( R e. V -> ( Base ` O ) = ( Base ` (mulGrp ` O ) ) )
33	29, 32	eqtrd 2222	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` O ) ) )
34	33	eleq2d 2259	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` O ) ) ) )
35	22, 4	mgpplusgg 13295	. . . . . . . . . . 11 |- ( O e. _V -> ( .r ` O ) = ( +g ` (mulGrp ` O ) ) )
36	21, 35	syl 14	. . . . . . . . . 10 |- ( R e. V -> ( .r ` O ) = ( +g ` (mulGrp ` O ) ) )
37	36	oveqd 5914	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` O ) y ) = ( x ( +g ` (mulGrp ` O ) ) y ) )
38	37	eqeq1d 2198	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` O ) y ) = y <-> ( x ( +g ` (mulGrp ` O ) ) y ) = y ) )
39	36	oveqd 5914	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` O ) x ) = ( y ( +g ` (mulGrp ` O ) ) x ) )
40	39	eqeq1d 2198	. . . . . . . 8 |- ( R e. V -> ( ( y ( .r ` O ) x ) = y <-> ( y ( +g ` (mulGrp ` O ) ) x ) = y ) )
41	38, 40	anbi12d 473	. . . . . . 7 |- ( R e. V -> ( ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) <-> ( ( x ( +g ` (mulGrp ` O ) ) y ) = y /\ ( y ( +g ` (mulGrp ` O ) ) x ) = y ) ) )
42	33, 41	raleqbidv 2698	. . . . . 6 |- ( R e. V -> ( A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) <-> A. y e. ( Base ` (mulGrp ` O ) ) ( ( x ( +g ` (mulGrp ` O ) ) y ) = y /\ ( y ( +g ` (mulGrp ` O ) ) x ) = y ) ) )
43	34, 42	anbi12d 473	. . . . 5 |- ( R e. V -> ( ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) <-> ( x e. ( Base ` (mulGrp ` O ) ) /\ A. y e. ( Base ` (mulGrp ` O ) ) ( ( x ( +g ` (mulGrp ` O ) ) y ) = y /\ ( y ( +g ` (mulGrp ` O ) ) x ) = y ) ) ) )
44	43	iotabidv 5218	. . . 4 |- ( R e. V -> ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) ) = ( iota x ( x e. ( Base ` (mulGrp ` O ) ) /\ A. y e. ( Base ` (mulGrp ` O ) ) ( ( x ( +g ` (mulGrp ` O ) ) y ) = y /\ ( y ( +g ` (mulGrp ` O ) ) x ) = y ) ) ) )
45	28, 44	eqtr4d 2225	. . 3 |- ( R e. V -> ( 0g ` (mulGrp ` O ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` O ) y ) = y /\ ( y ( .r ` O ) x ) = y ) ) ) )
46		eqid 2189	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
47	46	mgpex 13296	. . . . 5 |- ( R e. V -> (mulGrp ` R ) e. _V )
48		eqid 2189	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
49		eqid 2189	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
50		eqid 2189	. . . . . 6 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
51	48, 49, 50	grpidvalg 12852	. . . . 5 |- ( (mulGrp ` R ) e. _V -> ( 0g ` (mulGrp ` R ) ) = ( iota x ( x e. ( Base ` (mulGrp ` R ) ) /\ A. y e. ( Base ` (mulGrp ` R ) ) ( ( x ( +g ` (mulGrp ` R ) ) y ) = y /\ ( y ( +g ` (mulGrp ` R ) ) x ) = y ) ) ) )
52	47, 51	syl 14	. . . 4 |- ( R e. V -> ( 0g ` (mulGrp ` R ) ) = ( iota x ( x e. ( Base ` (mulGrp ` R ) ) /\ A. y e. ( Base ` (mulGrp ` R ) ) ( ( x ( +g ` (mulGrp ` R ) ) y ) = y /\ ( y ( +g ` (mulGrp ` R ) ) x ) = y ) ) ) )
53	46, 1	mgpbasg 13297	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
54	53	eleq2d 2259	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` R ) ) ) )
55	46, 2	mgpplusgg 13295	. . . . . . . . . 10 |- ( R e. V -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
56	55	oveqd 5914	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` R ) y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
57	56	eqeq1d 2198	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` R ) y ) = y <-> ( x ( +g ` (mulGrp ` R ) ) y ) = y ) )
58	55	oveqd 5914	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` R ) x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
59	58	eqeq1d 2198	. . . . . . . 8 |- ( R e. V -> ( ( y ( .r ` R ) x ) = y <-> ( y ( +g ` (mulGrp ` R ) ) x ) = y ) )
60	57, 59	anbi12d 473	. . . . . . 7 |- ( R e. V -> ( ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) <-> ( ( x ( +g ` (mulGrp ` R ) ) y ) = y /\ ( y ( +g ` (mulGrp ` R ) ) x ) = y ) ) )
61	53, 60	raleqbidv 2698	. . . . . 6 |- ( R e. V -> ( A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) <-> A. y e. ( Base ` (mulGrp ` R ) ) ( ( x ( +g ` (mulGrp ` R ) ) y ) = y /\ ( y ( +g ` (mulGrp ` R ) ) x ) = y ) ) )
62	54, 61	anbi12d 473	. . . . 5 |- ( R e. V -> ( ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) <-> ( x e. ( Base ` (mulGrp ` R ) ) /\ A. y e. ( Base ` (mulGrp ` R ) ) ( ( x ( +g ` (mulGrp ` R ) ) y ) = y /\ ( y ( +g ` (mulGrp ` R ) ) x ) = y ) ) ) )
63	62	iotabidv 5218	. . . 4 |- ( R e. V -> ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) ) = ( iota x ( x e. ( Base ` (mulGrp ` R ) ) /\ A. y e. ( Base ` (mulGrp ` R ) ) ( ( x ( +g ` (mulGrp ` R ) ) y ) = y /\ ( y ( +g ` (mulGrp ` R ) ) x ) = y ) ) ) )
64	52, 63	eqtr4d 2225	. . 3 |- ( R e. V -> ( 0g ` (mulGrp ` R ) ) = ( iota x ( x e. ( Base ` R ) /\ A. y e. ( Base ` R ) ( ( x ( .r ` R ) y ) = y /\ ( y ( .r ` R ) x ) = y ) ) ) )
65	20, 45, 64	3eqtr4d 2232	. 2 |- ( R e. V -> ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` R ) ) )
66		eqid 2189	. . . 4 |- ( 1r ` O ) = ( 1r ` O )
67	22, 66	ringidvalg 13332	. . 3 |- ( O e. _V -> ( 1r ` O ) = ( 0g ` (mulGrp ` O ) ) )
68	21, 67	syl 14	. 2 |- ( R e. V -> ( 1r ` O ) = ( 0g ` (mulGrp ` O ) ) )
69		oppr1.2	. . 3 |- .1. = ( 1r ` R )
70	46, 69	ringidvalg 13332	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
71	65, 68, 70	3eqtr4rd 2233	1 |- ( R e. V -> .1. = ( 1r ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2160   A.wral 2468   _Vcvv 2752   iotacio 5194   ` cfv 5235   iota_crio 5851  (class class class)co 5897   Basecbs 12515   +g cplusg 12592   .rcmulr 12593   0gc0g 12764  mulGrpcmgp 13291   1rcur 13330  opp_rcoppr 13434

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-i2m1 7947  ax-0lt1 7948  ax-0id 7950  ax-rnegex 7951  ax-pre-ltirr 7954  ax-pre-lttrn 7956  ax-pre-ltadd 7958

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-tpos 6271  df-pnf 8025  df-mnf 8026  df-ltxr 8028  df-inn 8951  df-2 9009  df-3 9010  df-ndx 12518  df-slot 12519  df-base 12521  df-sets 12522  df-plusg 12605  df-mulr 12606  df-0g 12766  df-mgp 13292  df-ur 13331  df-oppr 13435

This theorem is referenced by:  opprunitd  13477  rhmopp  13543