Theorem mgpval 18492

Description: Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)

Hypotheses

Ref	Expression
mgpval.1	|- M = (mulGrp ` R )
mgpval.2	|- .x. = ( .r ` R )

Assertion

Ref	Expression
mgpval	|- M = ( R sSet <. ( +g ` ndx ) , .x. >. )

Proof of Theorem mgpval

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgpval.1	. 2 |- M = (mulGrp ` R )
2		id 22	. . . . 5 |- ( r = R -> r = R )
3		fveq2 6191	. . . . . . 7 |- ( r = R -> ( .r ` r ) = ( .r ` R ) )
4		mgpval.2	. . . . . . 7 |- .x. = ( .r ` R )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( r = R -> ( .r ` r ) = .x. )
6	5	opeq2d 4409	. . . . 5 |- ( r = R -> <. ( +g ` ndx ) , ( .r ` r ) >. = <. ( +g ` ndx ) , .x. >. )
7	2, 6	oveq12d 6668	. . . 4 |- ( r = R -> ( r sSet <. ( +g ` ndx ) , ( .r ` r ) >. ) = ( R sSet <. ( +g ` ndx ) , .x. >. ) )
8		df-mgp 18490	. . . 4 |- mulGrp = ( r e. _V |-> ( r sSet <. ( +g ` ndx ) , ( .r ` r ) >. ) )
9		ovex 6678	. . . 4 |- ( R sSet <. ( +g ` ndx ) , .x. >. ) e. _V
10	7, 8, 9	fvmpt 6282	. . 3 |- ( R e. _V -> (mulGrp ` R ) = ( R sSet <. ( +g ` ndx ) , .x. >. ) )
11		fvprc 6185	. . . 4 |- ( -. R e. _V -> (mulGrp ` R ) = (/) )
12		reldmsets 15886	. . . . 5 |- Rel dom sSet
13	12	ovprc1 6684	. . . 4 |- ( -. R e. _V -> ( R sSet <. ( +g ` ndx ) , .x. >. ) = (/) )
14	11, 13	eqtr4d 2659	. . 3 |- ( -. R e. _V -> (mulGrp ` R ) = ( R sSet <. ( +g ` ndx ) , .x. >. ) )
15	10, 14	pm2.61i 176	. 2 |- (mulGrp ` R ) = ( R sSet <. ( +g ` ndx ) , .x. >. )
16	1, 15	eqtri 2644	1 |- M = ( R sSet <. ( +g ` ndx ) , .x. >. )

Syntax hints:   -. wn 3   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   <.cop 4183   ` cfv 5888  (class class class)co 6650   ndxcnx 15854   sSet csts 15855   +g cplusg 15941   .rcmulr 15942  mulGrpcmgp 18489

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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sets 15864  df-mgp 18490

This theorem is referenced by:  mgpplusg  18493  mgplem  18494  mgpress  18500