Theorem grpodivfval 27388

Description: Group division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
grpdiv.1	|- X = ran G
grpdiv.2	|- N = ( inv ` G )
grpdiv.3	|- D = ( /g ` G )

Assertion

Ref	Expression
grpodivfval	|- ( G e. GrpOp -> D = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )

Distinct variable groups:   x, y, G   x, N, y   x, X, y

Allowed substitution hints:   D( x, y)

Proof of Theorem grpodivfval

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpdiv.3	. 2 |- D = ( /g ` G )
2		grpdiv.1	. . . . 5 |- X = ran G
3		rnexg 7098	. . . . 5 |- ( G e. GrpOp -> ran G e. _V )
4	2, 3	syl5eqel 2705	. . . 4 |- ( G e. GrpOp -> X e. _V )
5		mpt2exga 7246	. . . 4 |- ( ( X e. _V /\ X e. _V ) -> ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) e. _V )
6	4, 4, 5	syl2anc 693	. . 3 |- ( G e. GrpOp -> ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) e. _V )
7		rneq 5351	. . . . . 6 |- ( g = G -> ran g = ran G )
8	7, 2	syl6eqr 2674	. . . . 5 |- ( g = G -> ran g = X )
9		id 22	. . . . . 6 |- ( g = G -> g = G )
10		eqidd 2623	. . . . . 6 |- ( g = G -> x = x )
11		fveq2 6191	. . . . . . . 8 |- ( g = G -> ( inv ` g ) = ( inv ` G ) )
12		grpdiv.2	. . . . . . . 8 |- N = ( inv ` G )
13	11, 12	syl6eqr 2674	. . . . . . 7 |- ( g = G -> ( inv ` g ) = N )
14	13	fveq1d 6193	. . . . . 6 |- ( g = G -> ( ( inv ` g ) ` y ) = ( N ` y ) )
15	9, 10, 14	oveq123d 6671	. . . . 5 |- ( g = G -> ( x g ( ( inv ` g ) ` y ) ) = ( x G ( N ` y ) ) )
16	8, 8, 15	mpt2eq123dv 6717	. . . 4 |- ( g = G -> ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )
17		df-gdiv 27350	. . . 4 |- /g = ( g e. GrpOp |-> ( x e. ran g , y e. ran g |-> ( x g ( ( inv ` g ) ` y ) ) ) )
18	16, 17	fvmptg 6280	. . 3 |- ( ( G e. GrpOp /\ ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) e. _V ) -> ( /g ` G ) = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )
19	6, 18	mpdan 702	. 2 |- ( G e. GrpOp -> ( /g ` G ) = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )
20	1, 19	syl5eq 2668	1 |- ( G e. GrpOp -> D = ( x e. X , y e. X |-> ( x G ( N ` y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   GrpOpcgr 27343   invcgn 27345   /g cgs 27346

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

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-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-gdiv 27350

This theorem is referenced by:  grpodivval  27389  grpodivf  27392  nvmfval  27499