Theorem lsmdisj2a

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	lsmdisj2a	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$

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	2	adantr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\in \mathrm{SubGrp}\left(G\right)$$
7	3	adantr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\in \mathrm{SubGrp}\left(G\right)$$
8	4	adantr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to U\in \mathrm{SubGrp}\left(G\right)$$
9		simprl	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}$$
10		simprr	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap T=\left\{\underset{\dot{}}{0}\right\}$$
11	1 6 7 8 5 9 10	lsmdisj2	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$
12	1 6 7 8 5 9	lsmdisj	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)$$
13	12	simpld	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap U=\left\{\underset{\dot{}}{0}\right\}$$
14	11 13	jca	$$⊢\left(\phi \wedge \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)$$
15		incom	$$⊢\left(S\underset{\dot{}}{\oplus }T\right)\cap U=U\cap \left(S\underset{\dot{}}{\oplus }T\right)$$
16	2	adantr	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\in \mathrm{SubGrp}\left(G\right)$$
17	4	adantr	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to U\in \mathrm{SubGrp}\left(G\right)$$
18	3	adantr	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\in \mathrm{SubGrp}\left(G\right)$$
19		incom	$$⊢\left(S\underset{\dot{}}{\oplus }U\right)\cap T=T\cap \left(S\underset{\dot{}}{\oplus }U\right)$$
20		simprl	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}$$
21	19 20	syl5eq	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\underset{\dot{}}{\oplus }U\right)\cap T=\left\{\underset{\dot{}}{0}\right\}$$
22		simprr	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap U=\left\{\underset{\dot{}}{0}\right\}$$
23	1 16 17 18 5 21 22	lsmdisj2	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to U\cap \left(S\underset{\dot{}}{\oplus }T\right)=\left\{\underset{\dot{}}{0}\right\}$$
24	15 23	syl5eq	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}$$
25		incom	$$⊢S\cap T=T\cap S$$
26	1 18 16 17 5 20	lsmdisjr	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(T\cap S=\left\{\underset{\dot{}}{0}\right\}\wedge T\cap U=\left\{\underset{\dot{}}{0}\right\}\right)$$
27	26	simpld	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to T\cap S=\left\{\underset{\dot{}}{0}\right\}$$
28	25 27	syl5eq	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to S\cap T=\left\{\underset{\dot{}}{0}\right\}$$
29	24 28	jca	$$⊢\left(\phi \wedge \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)\to \left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)$$
30	14 29	impbida	$$⊢\phi \to \left(\left(\left(S\underset{\dot{}}{\oplus }T\right)\cap U=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap T=\left\{\underset{\dot{}}{0}\right\}\right)\leftrightarrow \left(T\cap \left(S\underset{\dot{}}{\oplus }U\right)=\left\{\underset{\dot{}}{0}\right\}\wedge S\cap U=\left\{\underset{\dot{}}{0}\right\}\right)\right)$$