Theorem oppr1g 14176

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 2231	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
2		eqid 2231	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
3		opprbas.1	. . . . . . . . . . 11 |- O = (oppr ` R )
4		eqid 2231	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
5	1, 2, 3, 4	opprmulg 14165	. . . . . . . . . 10 |- ( ( R e. V /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
6	5	3expa 1230	. . . . . . . . 9 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
7	6	eqeq1d 2240	. . . . . . . 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 529	. . . . . . . . . 10 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> x e. ( Base ` R ) )
11	1, 2, 3, 4	opprmulg 14165	. . . . . . . . . 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 1274	. . . . . . . . 9 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
13	12	eqeq1d 2240	. . . . . . . 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 2529	. . . . 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 5999	. . . 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 5981	. . . 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 5981	. . . 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 2287	. . 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 14167	. . . . 5 |- ( R e. V -> O e. _V )
22		eqid 2231	. . . . . 6 |- (mulGrp ` O ) = (mulGrp ` O )
23	22	mgpex 14019	. . . . 5 |- ( O e. _V -> (mulGrp ` O ) e. _V )
24		eqid 2231	. . . . . 6 |- ( Base ` (mulGrp ` O ) ) = ( Base ` (mulGrp ` O ) )
25		eqid 2231	. . . . . 6 |- ( +g ` (mulGrp ` O ) ) = ( +g ` (mulGrp ` O ) )
26		eqid 2231	. . . . . 6 |- ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` O ) )
27	24, 25, 26	grpidvalg 13536	. . . . 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 14169	. . . . . . . 8 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
30		eqid 2231	. . . . . . . . . 10 |- ( Base ` O ) = ( Base ` O )
31	22, 30	mgpbasg 14020	. . . . . . . . 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 2264	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` O ) ) )
34	33	eleq2d 2301	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` O ) ) ) )
35	22, 4	mgpplusgg 14018	. . . . . . . . . . 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 6045	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` O ) y ) = ( x ( +g ` (mulGrp ` O ) ) y ) )
38	37	eqeq1d 2240	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` O ) y ) = y <-> ( x ( +g ` (mulGrp ` O ) ) y ) = y ) )
39	36	oveqd 6045	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` O ) x ) = ( y ( +g ` (mulGrp ` O ) ) x ) )
40	39	eqeq1d 2240	. . . . . . . 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 2747	. . . . . 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 5316	. . . 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 2267	. . 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 2231	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
47	46	mgpex 14019	. . . . 5 |- ( R e. V -> (mulGrp ` R ) e. _V )
48		eqid 2231	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
49		eqid 2231	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
50		eqid 2231	. . . . . 6 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
51	48, 49, 50	grpidvalg 13536	. . . . 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 14020	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
54	53	eleq2d 2301	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` R ) ) ) )
55	46, 2	mgpplusgg 14018	. . . . . . . . . 10 |- ( R e. V -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
56	55	oveqd 6045	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` R ) y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
57	56	eqeq1d 2240	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` R ) y ) = y <-> ( x ( +g ` (mulGrp ` R ) ) y ) = y ) )
58	55	oveqd 6045	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` R ) x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
59	58	eqeq1d 2240	. . . . . . . 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 2747	. . . . . 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 5316	. . . 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 2267	. . 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 2274	. 2 |- ( R e. V -> ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` R ) ) )
66		eqid 2231	. . . 4 |- ( 1r ` O ) = ( 1r ` O )
67	22, 66	ringidvalg 14055	. . 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 14055	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
71	65, 68, 70	3eqtr4rd 2275	1 |- ( R e. V -> .1. = ( 1r ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   iotacio 5291   ` cfv 5333   iota_crio 5980  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   0gc0g 13419  mulGrpcmgp 14014   1rcur 14053  opp_rcoppr 14161

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-lttrn 8206  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-tpos 6454  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgp 14015  df-ur 14054  df-oppr 14162

This theorem is referenced by:  opprunitd  14205  rhmopp  14271  opprnzrbg  14280