Theorem lsmdisj3

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	lsmcntz.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
	lsmcntz.s	$$⊢\phi \to S\in \mathrm{SubGrp}\left(G\right)$$
	lsmcntz.t	$$⊢\phi \to T\in \mathrm{SubGrp}\left(G\right)$$
	lsmcntz.u	$$⊢\phi \to U\in \mathrm{SubGrp}\left(G\right)$$
	lsmdisj.o	$$⊢\underset{\dot{}}{0}=0_G$$
	lsmdisj.i	$$⊢\phi \to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}$$
	lsmdisj2.i	$$⊢\phi \to S\cap T=\left\{\underset{\dot{}}{0}\right\}$$
	lsmdisj3.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
	lsmdisj3.s	$$⊢\phi \to S\subseteq Z\left(T\right)$$
Assertion	lsmdisj3	$$⊢\phi \to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$

Proof

Step	Hyp	Ref	Expression
1		lsmcntz.p	$$⊢\underset{\dot{}}{\oplus }=\mathrm{LSSum}\left(G\right)$$
2		lsmcntz.s	$$⊢\phi \to S\in \mathrm{SubGrp}\left(G\right)$$
3		lsmcntz.t	$$⊢\phi \to T\in \mathrm{SubGrp}\left(G\right)$$
4		lsmcntz.u	$$⊢\phi \to U\in \mathrm{SubGrp}\left(G\right)$$
5		lsmdisj.o	$$⊢\underset{\dot{}}{0}=0_G$$
6		lsmdisj.i	$$⊢\phi \to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}$$
7		lsmdisj2.i	$$⊢\phi \to S\cap T=\left\{\underset{\dot{}}{0}\right\}$$
8		lsmdisj3.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
9		lsmdisj3.s	$$⊢\phi \to S\subseteq Z\left(T\right)$$
10	1 8	lsmcom2	$$⊢\left(S\in \mathrm{SubGrp}\left(G\right)\wedge T\in \mathrm{SubGrp}\left(G\right)\wedge S\subseteq Z\left(T\right)\right)\to S\underset{\dot{}}{\oplus }T=T\underset{\dot{}}{\oplus }S$$
11	2 3 9 10	syl3anc	$$⊢\phi \to S\underset{\dot{}}{\oplus }T=T\underset{\dot{}}{\oplus }S$$
12	11	ineq1d	$$⊢\phi \to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left(T\underset{\dot{}}{\oplus }S\right)\cap U$$
13	12 6	eqtr3d	$$⊢\phi \to \left(T\underset{\dot{}}{\oplus }S\right)\cap U=\left\{\underset{\dot{}}{0}\right\}$$
14		incom	$$⊢T\cap S=S\cap T$$
15	14 7	syl5eq	$$⊢\phi \to T\cap S=\left\{\underset{\dot{}}{0}\right\}$$
16	1 3 2 4 5 13 15	lsmdisj2	$$⊢\phi \to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$