Theorem lsmdisj3b

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	$$⊢\underset{\dot{}}{0}=0_G$$
	lsmdisj3b.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
	lsmdisj3b.2	$$⊢\phi \to T\subseteq Z\left(U\right)$$
Assertion	lsmdisj3b	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$

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		lsmdisj3b.z	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
7		lsmdisj3b.2	$$⊢\phi \to T\subseteq Z\left(U\right)$$
8	1 2 4 3 5	lsmdisj2b	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(U\underset{\dot{}}{\oplus }T\right)=\left\{\underset{\dot{}}{0}\right\}\wedge U\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$
9	1 6	lsmcom2	$$⊢\left(T\in \mathrm{SubGrp}\left(G\right)\wedge U\in \mathrm{SubGrp}\left(G\right)\wedge T\subseteq Z\left(U\right)\right)\to T\underset{\dot{}}{\oplus }U=U\underset{\dot{}}{\oplus }T$$
10	3 4 7 9	syl3anc	$$⊢\phi \to T\underset{\dot{}}{\oplus }U=U\underset{\dot{}}{\oplus }T$$
11	10	ineq2d	$$⊢\phi \to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=S\cap \left(U\underset{\dot{}}{\oplus }T\right)$$
12	11	eqeq1d	$$⊢\phi \to \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\leftrightarrow S\cap \left(U\underset{\dot{}}{\oplus }T\right)=\left\{\underset{\dot{}}{0}\right\}\right)$$
13		incom	$$⊢T\cap U=U\cap T$$
14	13	a1i	$$⊢\phi \to T\cap U=U\cap T$$
15	14	eqeq1d	$$⊢\phi \to \left(T\cap U=\left\{\underset{\dot{}}{0}\right\}\leftrightarrow U\cap T=\left\{\underset{\dot{}}{0}\right\}\right)$$
16	12 15	anbi12d	$$⊢\phi \to \left(\left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(U\underset{\dot{}}{\oplus }T\right)=\left\{\underset{\dot{}}{0}\right\}\wedge U\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$
17	8 16	bitr4d	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$