Theorem oppr0g 14093

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 14087	. . . . 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 14088	. . . . . . . 8 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
7	6	oveqd 6034	. . . . . . 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 6034	. . . . . . 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 2746	. . . 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 5309	. 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 13455	. 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 14085	. . 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 13455	. . 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 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   iotacio 5284   ` cfv 5326  (class class class)co 6017   Basecbs 13081   +g cplusg 13159   0gc0g 13338  opp_rcoppr 14079

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-tpos 6410  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-oppr 14080

This theorem is referenced by:  opprnegg  14095  opprnzrbg  14198  opprdomnbg  14287  ridl0  14523  2idlcpblrng  14536