Theorem oppr1g 14326

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 2234	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
2		eqid 2234	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
3		opprbas.1	. . . . . . . . . . 11 |- O = (oppr ` R )
4		eqid 2234	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
5	1, 2, 3, 4	opprmulg 14314	. . . . . . . . . 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 2243	. . . . . . . 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 14314	. . . . . . . . . 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 2243	. . . . . . . 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 2540	. . . . 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 6029	. . . 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 6011	. . . 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 6011	. . . 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 2290	. . 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 14316	. . . . 5 |- ( R e. V -> O e. _V )
22		eqid 2234	. . . . . 6 |- (mulGrp ` O ) = (mulGrp ` O )
23	22	mgpex 14164	. . . . 5 |- ( O e. _V -> (mulGrp ` O ) e. _V )
24		eqid 2234	. . . . . 6 |- ( Base ` (mulGrp ` O ) ) = ( Base ` (mulGrp ` O ) )
25		eqid 2234	. . . . . 6 |- ( +g ` (mulGrp ` O ) ) = ( +g ` (mulGrp ` O ) )
26		eqid 2234	. . . . . 6 |- ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` O ) )
27	24, 25, 26	grpidvalg 13636	. . . . 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 14318	. . . . . . . 8 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
30		eqid 2234	. . . . . . . . . 10 |- ( Base ` O ) = ( Base ` O )
31	22, 30	mgpbasg 14165	. . . . . . . . 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 2267	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` O ) ) )
34	33	eleq2d 2304	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` O ) ) ) )
35	22, 4	mgpplusgg 14163	. . . . . . . . . . 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 6075	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` O ) y ) = ( x ( +g ` (mulGrp ` O ) ) y ) )
38	37	eqeq1d 2243	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` O ) y ) = y <-> ( x ( +g ` (mulGrp ` O ) ) y ) = y ) )
39	36	oveqd 6075	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` O ) x ) = ( y ( +g ` (mulGrp ` O ) ) x ) )
40	39	eqeq1d 2243	. . . . . . . 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 2759	. . . . . 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 5340	. . . 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 2270	. . 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 2234	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
47	46	mgpex 14164	. . . . 5 |- ( R e. V -> (mulGrp ` R ) e. _V )
48		eqid 2234	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
49		eqid 2234	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
50		eqid 2234	. . . . . 6 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
51	48, 49, 50	grpidvalg 13636	. . . . 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 14165	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
54	53	eleq2d 2304	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` R ) ) ) )
55	46, 2	mgpplusgg 14163	. . . . . . . . . 10 |- ( R e. V -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
56	55	oveqd 6075	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` R ) y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
57	56	eqeq1d 2243	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` R ) y ) = y <-> ( x ( +g ` (mulGrp ` R ) ) y ) = y ) )
58	55	oveqd 6075	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` R ) x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
59	58	eqeq1d 2243	. . . . . . . 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 2759	. . . . . 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 5340	. . . 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 2270	. . 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 2277	. 2 |- ( R e. V -> ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` R ) ) )
66		eqid 2234	. . . 4 |- ( 1r ` O ) = ( 1r ` O )
67	22, 66	ringidvalg 14204	. . 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 14204	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
71	65, 68, 70	3eqtr4rd 2278	1 |- ( R e. V -> .1. = ( 1r ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   iotacio 5315   ` cfv 5357   iota_crio 6010  (class class class)co 6058   Basecbs 13296   +g cplusg 13374   .rcmulr 13375   0gc0g 13553  mulGrpcmgp 14159   1rcur 14202  opp_rcoppr 14310

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 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-lttrn 8257  ax-pre-ltadd 8259

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-tpos 6489  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgp 14160  df-ur 14203  df-oppr 14311

This theorem is referenced by:  opprunitd  14355  rhmopp  14421  opprnzrbg  14430  opprlring  14442