Theorem opprnegg 13878

Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
opprbas.1	|- O = (oppr ` R )
opprneg.2	|- N = ( invg ` R )

Assertion

Ref	Expression
opprnegg	|- ( R e. V -> N = ( invg ` O ) )

Proof of Theorem opprnegg

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

Step	Hyp	Ref	Expression
1		opprbas.1	. . . 4 |- O = (oppr ` R )
2		eqid 2205	. . . 4 |- ( Base ` R ) = ( Base ` R )
3	1, 2	opprbasg 13870	. . 3 |- ( R e. V -> ( Base ` R ) = ( Base ` O ) )
4		eqid 2205	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
5	1, 4	oppraddg 13871	. . . . . 6 |- ( R e. V -> ( +g ` R ) = ( +g ` O ) )
6	5	oveqd 5963	. . . . 5 |- ( R e. V -> ( y ( +g ` R ) x ) = ( y ( +g ` O ) x ) )
7		eqid 2205	. . . . . 6 |- ( 0g ` R ) = ( 0g ` R )
8	1, 7	oppr0g 13876	. . . . 5 |- ( R e. V -> ( 0g ` R ) = ( 0g ` O ) )
9	6, 8	eqeq12d 2220	. . . 4 |- ( R e. V -> ( ( y ( +g ` R ) x ) = ( 0g ` R ) <-> ( y ( +g ` O ) x ) = ( 0g ` O ) ) )
10	3, 9	riotaeqbidv 5904	. . 3 |- ( R e. V -> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) = ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) )
11	3, 10	mpteq12dv 4127	. 2 |- ( R e. V -> ( x e. ( Base ` R ) |-> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) ) = ( x e. ( Base ` O ) |-> ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) ) )
12		opprneg.2	. . 3 |- N = ( invg ` R )
13	2, 4, 7, 12	grpinvfvalg 13407	. 2 |- ( R e. V -> N = ( x e. ( Base ` R ) |-> ( iota_ y e. ( Base ` R ) ( y ( +g ` R ) x ) = ( 0g ` R ) ) ) )
14	1	opprex 13868	. . 3 |- ( R e. V -> O e. _V )
15		eqid 2205	. . . 4 |- ( Base ` O ) = ( Base ` O )
16		eqid 2205	. . . 4 |- ( +g ` O ) = ( +g ` O )
17		eqid 2205	. . . 4 |- ( 0g ` O ) = ( 0g ` O )
18		eqid 2205	. . . 4 |- ( invg ` O ) = ( invg ` O )
19	15, 16, 17, 18	grpinvfvalg 13407	. . 3 |- ( O e. _V -> ( invg ` O ) = ( x e. ( Base ` O ) |-> ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) ) )
20	14, 19	syl 14	. 2 |- ( R e. V -> ( invg ` O ) = ( x e. ( Base ` O ) |-> ( iota_ y e. ( Base ` O ) ( y ( +g ` O ) x ) = ( 0g ` O ) ) ) )
21	11, 13, 20	3eqtr4d 2248	1 |- ( R e. V -> N = ( invg ` O ) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2176   _Vcvv 2772   |-> cmpt 4106   ` cfv 5272   iota_crio 5900  (class class class)co 5946   Basecbs 12865   +g cplusg 12942   0gc0g 13121   invgcminusg 13366  opp_rcoppr 13862

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-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-addass 8029  ax-i2m1 8032  ax-0lt1 8033  ax-0id 8035  ax-rnegex 8036  ax-pre-ltirr 8039  ax-pre-lttrn 8041  ax-pre-ltadd 8043

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-reu 2491  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-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-tpos 6333  df-pnf 8111  df-mnf 8112  df-ltxr 8114  df-inn 9039  df-2 9097  df-3 9098  df-ndx 12868  df-slot 12869  df-base 12871  df-sets 12872  df-plusg 12955  df-mulr 12956  df-0g 13123  df-minusg 13369  df-oppr 13863

This theorem is referenced by:  unitnegcl  13925