Theorem lsmdisj3

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-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 )
	lsmdisj.i	|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )
	lsmdisj2.i	|- ( ph -> ( S i^i T ) = { .0. } )
	lsmdisj3.z	|- Z = ( Cntz ` G )
	lsmdisj3.s	|- ( ph -> S C_ ( Z ` T ) )
Assertion	lsmdisj3	|- ( ph -> ( S i^i ( T .(+) 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		lsmdisj.i	|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )
7		lsmdisj2.i	|- ( ph -> ( S i^i T ) = { .0. } )
8		lsmdisj3.z	|- Z = ( Cntz ` G )
9		lsmdisj3.s	|- ( ph -> S C_ ( Z ` T ) )
10	1 8	lsmcom2	|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ S C_ ( Z ` T ) ) -> ( S .(+) T ) = ( T .(+) S ) )
11	2 3 9 10	syl3anc	|- ( ph -> ( S .(+) T ) = ( T .(+) S ) )
12	11	ineq1d	|- ( ph -> ( ( S .(+) T ) i^i U ) = ( ( T .(+) S ) i^i U ) )
13	12 6	eqtr3d	|- ( ph -> ( ( T .(+) S ) i^i U ) = { .0. } )
14		incom	|- ( T i^i S ) = ( S i^i T )
15	14 7	eqtrid	|- ( ph -> ( T i^i S ) = { .0. } )
16	1 3 2 4 5 13 15	lsmdisj2	|- ( ph -> ( S i^i ( T .(+) U ) ) = { .0. } )