Theorem lsmdisj3r

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-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	$$⊢\underset{\dot{}}{0}=0_G$$
	lsmdisjr.i	$$⊢\phi \to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$
	lsmdisj2r.i	$$⊢\phi \to T\cap U=\left\{\underset{\dot{}}{0}\right\}$$
	lsmdisj3r.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
	lsmdisj3r.s	$$⊢\phi \to T\subseteq Z\left(U\right)$$
Assertion	lsmdisj3r	$$⊢\phi \to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}$$

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		lsmdisjr.i	$$⊢\phi \to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$
7		lsmdisj2r.i	$$⊢\phi \to T\cap U=\left\{\underset{\dot{}}{0}\right\}$$
8		lsmdisj3r.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
9		lsmdisj3r.s	$$⊢\phi \to T\subseteq Z\left(U\right)$$
10	1 8	lsmcom2	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\wedge T\subseteq Z\left(U\right)\right)\to T\underset{\dot{}}{\oplus }U=U\underset{\dot{}}{\oplus }T$$
11	3 4 9 10	syl3anc	$$⊢\phi \to T\underset{\dot{}}{\oplus }U=U\underset{\dot{}}{\oplus }T$$
12	11	ineq2d	$$⊢\phi \to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=S\cap \left(U\underset{\dot{}}{\oplus }T\right)$$
13	12 6	eqtr3d	$$⊢\phi \to S\cap \left(U\underset{\dot{}}{\oplus }T\right)=\left\{\underset{\dot{}}{0}\right\}$$
14		incom	$$⊢U\cap T=T\cap U$$
15	14 7	syl5eq	$$⊢\phi \to U\cap T=\left\{\underset{\dot{}}{0}\right\}$$
16	1 2 4 3 5 13 15	lsmdisj2r	$$⊢\phi \to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}$$