Theorem oppr1g 13844

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 2205	. . . . . . . . . . 11 |- ( Base ` R ) = ( Base ` R )
2		eqid 2205	. . . . . . . . . . 11 |- ( .r ` R ) = ( .r ` R )
3		opprbas.1	. . . . . . . . . . 11 |- O = (oppr ` R )
4		eqid 2205	. . . . . . . . . . 11 |- ( .r ` O ) = ( .r ` O )
5	1, 2, 3, 4	opprmulg 13833	. . . . . . . . . 10 |- ( ( R e. V /\ x e. ( Base ` R ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
6	5	3expa 1206	. . . . . . . . 9 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( x ( .r ` O ) y ) = ( y ( .r ` R ) x ) )
7	6	eqeq1d 2214	. . . . . . . 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 13833	. . . . . . . . . 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 1250	. . . . . . . . 9 |- ( ( ( R e. V /\ x e. ( Base ` R ) ) /\ y e. ( Base ` R ) ) -> ( y ( .r ` O ) x ) = ( x ( .r ` R ) y ) )
13	12	eqeq1d 2214	. . . . . . . 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 2502	. . . . 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 5916	. . . 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 5899	. . . 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 5899	. . . 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 2261	. . 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 13835	. . . . 5 |- ( R e. V -> O e. _V )
22		eqid 2205	. . . . . 6 |- (mulGrp ` O ) = (mulGrp ` O )
23	22	mgpex 13687	. . . . 5 |- ( O e. _V -> (mulGrp ` O ) e. _V )
24		eqid 2205	. . . . . 6 |- ( Base ` (mulGrp ` O ) ) = ( Base ` (mulGrp ` O ) )
25		eqid 2205	. . . . . 6 |- ( +g ` (mulGrp ` O ) ) = ( +g ` (mulGrp ` O ) )
26		eqid 2205	. . . . . 6 |- ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` O ) )
27	24, 25, 26	grpidvalg 13205	. . . . 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 13837	. . . . . . . 8 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
30		eqid 2205	. . . . . . . . . 10 |- ( Base ` O ) = ( Base ` O )
31	22, 30	mgpbasg 13688	. . . . . . . . 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 2238	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` O ) ) )
34	33	eleq2d 2275	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` O ) ) ) )
35	22, 4	mgpplusgg 13686	. . . . . . . . . . 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 5961	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` O ) y ) = ( x ( +g ` (mulGrp ` O ) ) y ) )
38	37	eqeq1d 2214	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` O ) y ) = y <-> ( x ( +g ` (mulGrp ` O ) ) y ) = y ) )
39	36	oveqd 5961	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` O ) x ) = ( y ( +g ` (mulGrp ` O ) ) x ) )
40	39	eqeq1d 2214	. . . . . . . 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 2718	. . . . . 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 5254	. . . 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 2241	. . 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 2205	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
47	46	mgpex 13687	. . . . 5 |- ( R e. V -> (mulGrp ` R ) e. _V )
48		eqid 2205	. . . . . 6 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
49		eqid 2205	. . . . . 6 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
50		eqid 2205	. . . . . 6 |- ( 0g ` (mulGrp ` R ) ) = ( 0g ` (mulGrp ` R ) )
51	48, 49, 50	grpidvalg 13205	. . . . 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 13688	. . . . . . 7 |- ( R e. V -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
54	53	eleq2d 2275	. . . . . 6 |- ( R e. V -> ( x e. ( Base ` R ) <-> x e. ( Base ` (mulGrp ` R ) ) ) )
55	46, 2	mgpplusgg 13686	. . . . . . . . . 10 |- ( R e. V -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
56	55	oveqd 5961	. . . . . . . . 9 |- ( R e. V -> ( x ( .r ` R ) y ) = ( x ( +g ` (mulGrp ` R ) ) y ) )
57	56	eqeq1d 2214	. . . . . . . 8 |- ( R e. V -> ( ( x ( .r ` R ) y ) = y <-> ( x ( +g ` (mulGrp ` R ) ) y ) = y ) )
58	55	oveqd 5961	. . . . . . . . 9 |- ( R e. V -> ( y ( .r ` R ) x ) = ( y ( +g ` (mulGrp ` R ) ) x ) )
59	58	eqeq1d 2214	. . . . . . . 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 2718	. . . . . 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 5254	. . . 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 2241	. . 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 2248	. 2 |- ( R e. V -> ( 0g ` (mulGrp ` O ) ) = ( 0g ` (mulGrp ` R ) ) )
66		eqid 2205	. . . 4 |- ( 1r ` O ) = ( 1r ` O )
67	22, 66	ringidvalg 13723	. . 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 13723	. 2 |- ( R e. V -> .1. = ( 0g ` (mulGrp ` R ) ) )
71	65, 68, 70	3eqtr4rd 2249	1 |- ( R e. V -> .1. = ( 1r ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2176   A.wral 2484   _Vcvv 2772   iotacio 5230   ` cfv 5271   iota_crio 5898  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   .rcmulr 12910   0gc0g 13088  mulGrpcmgp 13682   1rcur 13721  opp_rcoppr 13829

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-tpos 6331  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgp 13683  df-ur 13722  df-oppr 13830

This theorem is referenced by:  opprunitd  13872  rhmopp  13938  opprnzrbg  13947