Theorem lsmfval 18053

Description: The subgroup sum function (for a group or vector space). (Contributed by NM, 28-Jan-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)

Hypotheses

Ref	Expression
lsmfval.v	|- B = ( Base ` G )
lsmfval.a	|- .+ = ( +g ` G )
lsmfval.s	|- .(+) = ( LSSum ` G )

Assertion

Ref	Expression
lsmfval	|- ( G e. V -> .(+) = ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x .+ y ) ) ) )

Distinct variable groups:   u, t, x, y, .+   t, B, u, x, y   t, G, u, x, y

Allowed substitution hints:   .(+) ( x, y, u, t)   V( x, y, u, t)

Proof of Theorem lsmfval

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsmfval.s	. 2 |- .(+) = ( LSSum ` G )
2		elex 3212	. . 3 |- ( G e. V -> G e. _V )
3		fveq2 6191	. . . . . . 7 |- ( w = G -> ( Base ` w ) = ( Base ` G ) )
4		lsmfval.v	. . . . . . 7 |- B = ( Base ` G )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( w = G -> ( Base ` w ) = B )
6	5	pweqd 4163	. . . . 5 |- ( w = G -> ~P ( Base ` w ) = ~P B )
7		fveq2 6191	. . . . . . . . 9 |- ( w = G -> ( +g ` w ) = ( +g ` G ) )
8		lsmfval.a	. . . . . . . . 9 |- .+ = ( +g ` G )
9	7, 8	syl6eqr 2674	. . . . . . . 8 |- ( w = G -> ( +g ` w ) = .+ )
10	9	oveqd 6667	. . . . . . 7 |- ( w = G -> ( x ( +g ` w ) y ) = ( x .+ y ) )
11	10	mpt2eq3dv 6721	. . . . . 6 |- ( w = G -> ( x e. t , y e. u |-> ( x ( +g ` w ) y ) ) = ( x e. t , y e. u |-> ( x .+ y ) ) )
12	11	rneqd 5353	. . . . 5 |- ( w = G -> ran ( x e. t , y e. u |-> ( x ( +g ` w ) y ) ) = ran ( x e. t , y e. u |-> ( x .+ y ) ) )
13	6, 6, 12	mpt2eq123dv 6717	. . . 4 |- ( w = G -> ( t e. ~P ( Base ` w ) , u e. ~P ( Base ` w ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` w ) y ) ) ) = ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x .+ y ) ) ) )
14		df-lsm 18051	. . . 4 |- LSSum = ( w e. _V |-> ( t e. ~P ( Base ` w ) , u e. ~P ( Base ` w ) |-> ran ( x e. t , y e. u |-> ( x ( +g ` w ) y ) ) ) )
15		fvex 6201	. . . . . . 7 |- ( Base ` G ) e. _V
16	4, 15	eqeltri 2697	. . . . . 6 |- B e. _V
17	16	pwex 4848	. . . . 5 |- ~P B e. _V
18	17, 17	mpt2ex 7247	. . . 4 |- ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x .+ y ) ) ) e. _V
19	13, 14, 18	fvmpt 6282	. . 3 |- ( G e. _V -> ( LSSum ` G ) = ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x .+ y ) ) ) )
20	2, 19	syl 17	. 2 |- ( G e. V -> ( LSSum ` G ) = ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x .+ y ) ) ) )
21	1, 20	syl5eq 2668	1 |- ( G e. V -> .(+) = ( t e. ~P B , u e. ~P B |-> ran ( x e. t , y e. u |-> ( x .+ y ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   ~Pcpw 4158   ran crn 5115   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   +g cplusg 15941   LSSumclsm 18049

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-lsm 18051

This theorem is referenced by:  lsmvalx  18054  oppglsm  18057  lsmpropd  18090