Theorem oppr0g 14175

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 2231	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 14169	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
4	3	eleq2d 2301	. . . 4 |- ( R e. V -> ( y e. ( Base ` R ) <-> y e. ( Base ` O ) ) )
5		eqid 2231	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
6	1, 5	oppraddg 14170	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqd 6045	. . . . . . 7 |- ( R e. V -> ( y ( +g ` R ) x ) = ( y ( +g ` O ) x ) )
8	7	eqeq1d 2240	. . . . . 6 |- ( R e. V -> ( ( y ( +g ` R ) x ) = x <-> ( y ( +g ` O ) x ) = x ) )
9	6	oveqd 6045	. . . . . . 7 |- ( R e. V -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
10	9	eqeq1d 2240	. . . . . 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 2747	. . . 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 5316	. 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 13536	. 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 14167	. . 3 |- ( R e. V -> O e. _V )
18		eqid 2231	. . . 4 |- ( Base ` O ) = ( Base ` O )
19		eqid 2231	. . . 4 |- ( +g ` O ) = ( +g ` O )
20		eqid 2231	. . . 4 |- ( 0g ` O ) = ( 0g ` O )
21	18, 19, 20	grpidvalg 13536	. . 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 2274	1 |- ( R e. V -> .0. = ( 0g ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   _Vcvv 2803   iotacio 5291   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   0gc0g 13419  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-oppr 14162

This theorem is referenced by:  opprnegg  14177  opprnzrbg  14280  opprdomnbg  14370  ridl0  14606  2idlcpblrng  14619