Theorem oppgval 17777

Description: Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)

Hypotheses

Ref	Expression
oppgval.2	|- .+ = ( +g ` R )
oppgval.3	|- O = (oppg ` R )

Assertion

Ref	Expression
oppgval	|- O = ( R sSet <. ( +g ` ndx ) , tpos .+ >. )

Proof of Theorem oppgval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oppgval.3	. 2 |- O = (oppg ` R )
2		id 22	. . . . 5 |- ( x = R -> x = R )
3		fveq2 6191	. . . . . . . 8 |- ( x = R -> ( +g ` x ) = ( +g ` R ) )
4		oppgval.2	. . . . . . . 8 |- .+ = ( +g ` R )
5	3, 4	syl6eqr 2674	. . . . . . 7 |- ( x = R -> ( +g ` x ) = .+ )
6	5	tposeqd 7355	. . . . . 6 |- ( x = R -> tpos ( +g ` x ) = tpos .+ )
7	6	opeq2d 4409	. . . . 5 |- ( x = R -> <. ( +g ` ndx ) , tpos ( +g ` x ) >. = <. ( +g ` ndx ) , tpos .+ >. )
8	2, 7	oveq12d 6668	. . . 4 |- ( x = R -> ( x sSet <. ( +g ` ndx ) , tpos ( +g ` x ) >. ) = ( R sSet <. ( +g ` ndx ) , tpos .+ >. ) )
9		df-oppg 17776	. . . 4 |- oppg = ( x e. _V |-> ( x sSet <. ( +g ` ndx ) , tpos ( +g ` x ) >. ) )
10		ovex 6678	. . . 4 |- ( R sSet <. ( +g ` ndx ) , tpos .+ >. ) e. _V
11	8, 9, 10	fvmpt 6282	. . 3 |- ( R e. _V -> (oppg ` R ) = ( R sSet <. ( +g ` ndx ) , tpos .+ >. ) )
12		fvprc 6185	. . . 4 |- ( -. R e. _V -> (oppg ` R ) = (/) )
13		reldmsets 15886	. . . . 5 |- Rel dom sSet
14	13	ovprc1 6684	. . . 4 |- ( -. R e. _V -> ( R sSet <. ( +g ` ndx ) , tpos .+ >. ) = (/) )
15	12, 14	eqtr4d 2659	. . 3 |- ( -. R e. _V -> (oppg ` R ) = ( R sSet <. ( +g ` ndx ) , tpos .+ >. ) )
16	11, 15	pm2.61i 176	. 2 |- (oppg ` R ) = ( R sSet <. ( +g ` ndx ) , tpos .+ >. )
17	1, 16	eqtri 2644	1 |- O = ( R sSet <. ( +g ` ndx ) , tpos .+ >. )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650  tpos ctpos 7351   ndxcnx 15854   sSet csts 15855   +g cplusg 15941  opp_gcoppg 17775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-tpos 7352  df-sets 15864  df-oppg 17776

This theorem is referenced by:  oppgplusfval  17778  oppglem  17780