Theorem oppr0g 13256

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

Hypotheses

Ref	Expression
opprbas.1	|- O = (oppr ` R )
oppr0.2	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
oppr0g	|- ( R e. V -> .0. = ( 0g ` O ) )

Proof of Theorem oppr0g

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

Step	Hyp	Ref	Expression
1		opprbas.1	. . . . . 6 |- O = (oppr ` R )
2		eqid 2177	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 13252	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
4	3	eleq2d 2247	. . . 4 |- ( R e. V -> ( y e. ( Base ` R ) <-> y e. ( Base ` O ) ) )
5		eqid 2177	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
6	1, 5	oppraddg 13253	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqd 5894	. . . . . . 7 |- ( R e. V -> ( y ( +g ` R ) x ) = ( y ( +g ` O ) x ) )
8	7	eqeq1d 2186	. . . . . 6 |- ( R e. V -> ( ( y ( +g ` R ) x ) = x <-> ( y ( +g ` O ) x ) = x ) )
9	6	oveqd 5894	. . . . . . 7 |- ( R e. V -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
10	9	eqeq1d 2186	. . . . . 6 |- ( R e. V -> ( ( x ( +g ` R ) y ) = x <-> ( x ( +g ` O ) y ) = x ) )
11	8, 10	anbi12d 473	. . . . 5 |- ( R e. V -> ( ( ( y ( +g ` R ) x ) = x /\ ( x ( +g ` R ) y ) = x ) <-> ( ( y ( +g ` O ) x ) = x /\ ( x ( +g ` O ) y ) = x ) ) )
12	3, 11	raleqbidv 2685	. . . 4 |- ( R e. V -> ( A. x e. ( Base ` R ) ( ( y ( +g ` R ) x ) = x /\ ( x ( +g ` R ) y ) = x ) <-> A. x e. ( Base ` O ) ( ( y ( +g ` O ) x ) = x /\ ( x ( +g ` O ) y ) = x ) ) )
13	4, 12	anbi12d 473	. . 3 |- ( R e. V -> ( ( y e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( y ( +g ` R ) x ) = x /\ ( x ( +g ` R ) y ) = x ) ) <-> ( y e. ( Base ` O ) /\ A. x e. ( Base ` O ) ( ( y ( +g ` O ) x ) = x /\ ( x ( +g ` O ) y ) = x ) ) ) )
14	13	iotabidv 5201	. 2 |- ( R e. V -> ( iota y ( y e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( y ( +g ` R ) x ) = x /\ ( x ( +g ` R ) y ) = x ) ) ) = ( iota y ( y e. ( Base ` O ) /\ A. x e. ( Base ` O ) ( ( y ( +g ` O ) x ) = x /\ ( x ( +g ` O ) y ) = x ) ) ) )
15		oppr0.2	. . 3 |- .0. = ( 0g ` R )
16	2, 5, 15	grpidvalg 12797	. 2 |- ( R e. V -> .0. = ( iota y ( y e. ( Base ` R ) /\ A. x e. ( Base ` R ) ( ( y ( +g ` R ) x ) = x /\ ( x ( +g ` R ) y ) = x ) ) ) )
17	1	opprex 13250	. . 3 |- ( R e. V -> O e. _V )
18		eqid 2177	. . . 4 |- ( Base ` O ) = ( Base ` O )
19		eqid 2177	. . . 4 |- ( +g ` O ) = ( +g ` O )
20		eqid 2177	. . . 4 |- ( 0g ` O ) = ( 0g ` O )
21	18, 19, 20	grpidvalg 12797	. . 3 |- ( O e. _V -> ( 0g ` O ) = ( iota y ( y e. ( Base ` O ) /\ A. x e. ( Base ` O ) ( ( y ( +g ` O ) x ) = x /\ ( x ( +g ` O ) y ) = x ) ) ) )
22	17, 21	syl 14	. 2 |- ( R e. V -> ( 0g ` O ) = ( iota y ( y e. ( Base ` O ) /\ A. x e. ( Base ` O ) ( ( y ( +g ` O ) x ) = x /\ ( x ( +g ` O ) y ) = x ) ) ) )
23	14, 16, 22	3eqtr4d 2220	1 |- ( R e. V -> .0. = ( 0g ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2739   iotacio 5178   ` cfv 5218  (class class class)co 5877   Basecbs 12464   +g cplusg 12538   0gc0g 12710  opp_rcoppr 13244

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-pre-ltirr 7925  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-tpos 6248  df-pnf 7996  df-mnf 7997  df-ltxr 7999  df-inn 8922  df-2 8980  df-3 8981  df-ndx 12467  df-slot 12468  df-base 12470  df-sets 12471  df-plusg 12551  df-mulr 12552  df-0g 12712  df-oppr 13245

This theorem is referenced by:  opprnegg  13258