Theorem oppr0g 13928

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 2206	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 13922	. . . . 5 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
4	3	eleq2d 2276	. . . 4 |- ( R e. V -> ( y e. ( Base ` R ) <-> y e. ( Base ` O ) ) )
5		eqid 2206	. . . . . . . . 9 |- ( +g ` R ) = ( +g ` R )
6	1, 5	oppraddg 13923	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqd 5979	. . . . . . 7 |- ( R e. V -> ( y ( +g ` R ) x ) = ( y ( +g ` O ) x ) )
8	7	eqeq1d 2215	. . . . . 6 |- ( R e. V -> ( ( y ( +g ` R ) x ) = x <-> ( y ( +g ` O ) x ) = x ) )
9	6	oveqd 5979	. . . . . . 7 |- ( R e. V -> ( x ( +g ` R ) y ) = ( x ( +g ` O ) y ) )
10	9	eqeq1d 2215	. . . . . 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 2719	. . . 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 5268	. 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 13290	. 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 13920	. . 3 |- ( R e. V -> O e. _V )
18		eqid 2206	. . . 4 |- ( Base ` O ) = ( Base ` O )
19		eqid 2206	. . . 4 |- ( +g ` O ) = ( +g ` O )
20		eqid 2206	. . . 4 |- ( 0g ` O ) = ( 0g ` O )
21	18, 19, 20	grpidvalg 13290	. . 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 2249	1 |- ( R e. V -> .0. = ( 0g ` O ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2177   A.wral 2485   _Vcvv 2773   iotacio 5244   ` cfv 5285  (class class class)co 5962   Basecbs 12917   +g cplusg 12994   0gc0g 13173  opp_rcoppr 13914

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-addcom 8055  ax-addass 8057  ax-i2m1 8060  ax-0lt1 8061  ax-0id 8063  ax-rnegex 8064  ax-pre-ltirr 8067  ax-pre-lttrn 8069  ax-pre-ltadd 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-br 4055  df-opab 4117  df-mpt 4118  df-id 4353  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-tpos 6349  df-pnf 8139  df-mnf 8140  df-ltxr 8142  df-inn 9067  df-2 9125  df-3 9126  df-ndx 12920  df-slot 12921  df-base 12923  df-sets 12924  df-plusg 13007  df-mulr 13008  df-0g 13175  df-oppr 13915

This theorem is referenced by:  opprnegg  13930  opprnzrbg  14032  opprdomnbg  14121  ridl0  14357  2idlcpblrng  14370