Theorem lsmdisj2b

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	$$⊢\underset{\dot{}}{0}=0_G$$
Assertion	lsmdisj2b	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$

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		incom	$$⊢S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left(T\underset{\dot{}}{\oplus }U\right)\cap S$$
7	3	adantr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\in \mathrm{SubGrp}\left(G\right)$$
8	2	adantr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\in \mathrm{SubGrp}\left(G\right)$$
9	4	adantr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to U\in \mathrm{SubGrp}\left(G\right)$$
10		incom	$$⊢T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left(S\underset{\dot{}}{\oplus }U\right)\cap T$$
11		simprl	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}$$
12	10 11	syl5eq	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$
13		simprr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap U=\left\{\underset{\dot{}}{0}\right\}$$
14	1 7 8 9 5 12 13	lsmdisj2r	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(T\underset{\dot{}}{\oplus }U\right)\cap S=\left\{\underset{\dot{}}{0}\right\}$$
15	6 14	syl5eq	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$
16		incom	$$⊢T\cap U=U\cap T$$
17	1 8 9 7 5 11	lsmdisj	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge U\cap T=\left\{\underset{\dot{}}{0}\right\}\right)$$
18	17	simprd	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to U\cap T=\left\{\underset{\dot{}}{0}\right\}$$
19	16 18	syl5eq	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\cap U=\left\{\underset{\dot{}}{0}\right\}$$
20	15 19	jca	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)$$
21	2	adantr	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\in \mathrm{SubGrp}\left(G\right)$$
22	3	adantr	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\in \mathrm{SubGrp}\left(G\right)$$
23	4	adantr	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to U\in \mathrm{SubGrp}\left(G\right)$$
24		simprl	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$
25		simprr	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\cap U=\left\{\underset{\dot{}}{0}\right\}$$
26	1 21 22 23 5 24 25	lsmdisj2r	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}$$
27	1 21 22 23 5 24	lsmdisjr	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)$$
28	27	simprd	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap U=\left\{\underset{\dot{}}{0}\right\}$$
29	26 28	jca	$$⊢\left(\phi \wedge \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)$$
30	20 29	impbida	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(S\cap \left(T\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$