Theorem lsmdisj

Description: Disjointness from 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. } )
Assertion	lsmdisj	|- ( ph -> ( ( S i^i 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		lsmdisj.i	|- ( ph -> ( ( S .(+) T ) i^i U ) = { .0. } )
7	1	lsmub1	|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) -> S C_ ( S .(+) T ) )
8	2 3 7	syl2anc	|- ( ph -> S C_ ( S .(+) T ) )
9	8	ssrind	|- ( ph -> ( S i^i U ) C_ ( ( S .(+) T ) i^i U ) )
10	9 6	sseqtrd	|- ( ph -> ( S i^i U ) C_ { .0. } )
11	5	subg0cl	|- ( S e. ( SubGrp ` G ) -> .0. e. S )
12	2 11	syl	|- ( ph -> .0. e. S )
13	5	subg0cl	|- ( U e. ( SubGrp ` G ) -> .0. e. U )
14	4 13	syl	|- ( ph -> .0. e. U )
15	12 14	elind	|- ( ph -> .0. e. ( S i^i U ) )
16	15	snssd	|- ( ph -> { .0. } C_ ( S i^i U ) )
17	10 16	eqssd	|- ( ph -> ( S i^i U ) = { .0. } )
18	1	lsmub2	|- ( ( S e. ( SubGrp ` G ) /\ T e. ( SubGrp ` G ) ) -> T C_ ( S .(+) T ) )
19	2 3 18	syl2anc	|- ( ph -> T C_ ( S .(+) T ) )
20	19	ssrind	|- ( ph -> ( T i^i U ) C_ ( ( S .(+) T ) i^i U ) )
21	20 6	sseqtrd	|- ( ph -> ( T i^i U ) C_ { .0. } )
22	5	subg0cl	|- ( T e. ( SubGrp ` G ) -> .0. e. T )
23	3 22	syl	|- ( ph -> .0. e. T )
24	23 14	elind	|- ( ph -> .0. e. ( T i^i U ) )
25	24	snssd	|- ( ph -> { .0. } C_ ( T i^i U ) )
26	21 25	eqssd	|- ( ph -> ( T i^i U ) = { .0. } )
27	17 26	jca	|- ( ph -> ( ( S i^i U ) = { .0. } /\ ( T i^i U ) = { .0. } ) )