Theorem lsmdisj3a

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 )
	lsmdisj3b.z	|- Z = ( Cntz ` G )
	lsmdisj3a.2	|- ( ph -> S C_ ( Z ` T ) )
Assertion	lsmdisj3a	|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( 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		lsmdisj3b.z	|- Z = ( Cntz ` G )
7		lsmdisj3a.2	|- ( ph -> S C_ ( Z ` T ) )
8	1 6	lsmcom2	|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) /\ S C_ ( Z ` T ) ) -> ( S .(+) T ) = ( T .(+) S ) )
9	2 3 7 8	syl3anc	|- ( ph -> ( S .(+) T ) = ( T .(+) S ) )
10	9	ineq1d	|- ( ph -> ( ( S .(+) T ) i^i U ) = ( ( T .(+) S ) i^i U ) )
11	10	eqeq1d	|- ( ph -> ( ( ( S .(+) T ) i^i U ) = { .0. } <-> ( ( T .(+) S ) i^i U ) = { .0. } ) )
12		incom	|- ( S i^i T ) = ( T i^i S )
13	12	a1i	|- ( ph -> ( S i^i T ) = ( T i^i S ) )
14	13	eqeq1d	|- ( ph -> ( ( S i^i T ) = { .0. } <-> ( T i^i S ) = { .0. } ) )
15	11 14	anbi12d	|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( ( T .(+) S ) i^i U ) = { .0. } /\ ( T i^i S ) = { .0. } ) ) )
16	1 3 2 4 5	lsmdisj2a	|- ( ph -> ( ( ( ( T .(+) S ) i^i U ) = { .0. } /\ ( T i^i S ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) )
17	15 16	bitrd	|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) )