Theorem lsmdisj3a

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	$$⊢Z=\mathrm{Cntz}\left(G\right)$$
	lsmdisj3a.2	$$⊢\phi \to S\subseteq Z\left(T\right)$$
Assertion	lsmdisj3a	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$

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		lsmdisj3a.2	$$⊢\phi \to S\subseteq Z\left(T\right)$$
8	1 6	lsmcom2	$$⊢\left(S\in \mathrm{SubGrp}\left(G\right)\wedge T\in \mathrm{SubGrp}\left(G\right)\wedge S\subseteq Z\left(T\right)\right)\to S\underset{\dot{}}{\oplus }T=T\underset{\dot{}}{\oplus }S$$
9	2 3 7 8	syl3anc	$$⊢\phi \to S\underset{\dot{}}{\oplus }T=T\underset{\dot{}}{\oplus }S$$
10	9	ineq1d	$$⊢\phi \to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left(T\underset{\dot{}}{\oplus }S\right)\cap U$$
11	10	eqeq1d	$$⊢\phi \to \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\leftrightarrow \left(T\underset{\dot{}}{\oplus }S\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\right)$$
12		incom	$$⊢S\cap T=T\cap S$$
13	12	a1i	$$⊢\phi \to S\cap T=T\cap S$$
14	13	eqeq1d	$$⊢\phi \to \left(S\cap T=\left\{\underset{\dot{}}{0}\right\}\leftrightarrow T\cap S=\left\{\underset{\dot{}}{0}\right\}\right)$$
15	11 14	anbi12d	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(\left(T\underset{\dot{}}{\oplus }S\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap S=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$
16	1 3 2 4 5	lsmdisj2a	$$⊢\phi \to \left(\left(\left(T\underset{\dot{}}{\oplus }S\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap S=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$
17	15 16	bitrd	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$