Theorem lsmdisj3r

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016)

	Ref	Expression
Hypotheses	lsmcntz.p	|- .(+) = ( LSSum ` G )
	lsmcntz.s	|- ( ph -> S e. ( SubGrp ` G ) )
	lsmcntz.t	|- ( ph -> T e. ( SubGrp ` G ) )
	lsmcntz.u	|- ( ph -> U e. ( SubGrp ` G ) )
	lsmdisj.o	|- .0. = ( 0g ` G )
	lsmdisjr.i	|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
	lsmdisj2r.i	|- ( ph -> ( T i^i U ) = { .0. } )
	lsmdisj3r.z	|- Z = ( Cntz ` G )
	lsmdisj3r.s	|- ( ph -> T C_ ( Z ` U ) )
Assertion	lsmdisj3r	|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )

Proof

Step	Hyp	Ref	Expression
1		lsmcntz.p	|- .(+) = ( LSSum ` G )
2		lsmcntz.s	|- ( ph -> S e. ( SubGrp ` G ) )
3		lsmcntz.t	|- ( ph -> T e. ( SubGrp ` G ) )
4		lsmcntz.u	|- ( ph -> U e. ( SubGrp ` G ) )
5		lsmdisj.o	|- .0. = ( 0g ` G )
6		lsmdisjr.i	|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )
7		lsmdisj2r.i	|- ( ph -> ( T i^i U ) = { .0. } )
8		lsmdisj3r.z	|- Z = ( Cntz ` G )
9		lsmdisj3r.s	|- ( ph -> T C_ ( Z ` U ) )
10	1 8	lsmcom2	|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
11	3 4 9 10	syl3anc	|- ( ph -> ( T .(+) U ) = ( U .(+) T ) )
12	11	ineq2d	|- ( ph -> ( S i^i ( T .(+) U ) ) = ( S i^i ( U .(+) T ) ) )
13	12 6	eqtr3d	|- ( ph -> ( S i^i ( U .(+) T ) ) = { .0. } )
14		incom	|- ( U i^i T ) = ( T i^i U )
15	14 7	syl5eq	|- ( ph -> ( U i^i T ) = { .0. } )
16	1 2 4 3 5 13 15	lsmdisj2r	|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )