Theorem lsmdisj3b

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 )
	lsmdisj3b.2	|- ( ph -> T C_ ( Z ` U ) )
Assertion	lsmdisj3b	|- ( 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		lsmdisj3b.2	|- ( ph -> T C_ ( Z ` U ) )
8	1 2 4 3 5	lsmdisj2b	|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( S i^i ( U .(+) T ) ) = { .0. } /\ ( U i^i T ) = { .0. } ) ) )
9	1 6	lsmcom2	|- ( ( T e. ( SubGrp ` G ) /\ U e. ( SubGrp ` G ) /\ T C_ ( Z ` U ) ) -> ( T .(+) U ) = ( U .(+) T ) )
10	3 4 7 9	syl3anc	|- ( ph -> ( T .(+) U ) = ( U .(+) T ) )
11	10	ineq2d	|- ( ph -> ( S i^i ( T .(+) U ) ) = ( S i^i ( U .(+) T ) ) )
12	11	eqeq1d	|- ( ph -> ( ( S i^i ( T .(+) U ) ) = { .0. } <-> ( S i^i ( U .(+) T ) ) = { .0. } ) )
13		incom	|- ( T i^i U ) = ( U i^i T )
14	13	a1i	|- ( ph -> ( T i^i U ) = ( U i^i T ) )
15	14	eqeq1d	|- ( ph -> ( ( T i^i U ) = { .0. } <-> ( U i^i T ) = { .0. } ) )
16	12 15	anbi12d	|- ( ph -> ( ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) <-> ( ( S i^i ( U .(+) T ) ) = { .0. } /\ ( U i^i T ) = { .0. } ) ) )
17	8 16	bitr4d	|- ( ph -> ( ( ( ( S .(+) T ) i^i U ) = { .0. } /\ ( S i^i T ) = { .0. } ) <-> ( ( S i^i ( T .(+) U ) ) = { .0. } /\ ( T i^i U ) = { .0. } ) ) )