Theorem lsmdisj

Description: Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016)

	Ref	Expression
Hypotheses	lsmcntz.p	⊢ ⊕ = ( LSSum ‘ 𝐺 )
	lsmcntz.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
	lsmcntz.t	⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
	lsmcntz.u	⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
	lsmdisj.o	⊢ 0 = ( 0g ‘ 𝐺 )
	lsmdisj.i	⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } )
Assertion	lsmdisj	⊢ ( 𝜑 → ( ( 𝑆 ∩ 𝑈 ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) )

Proof

Step	Hyp	Ref	Expression
1		lsmcntz.p	⊢ ⊕ = ( LSSum ‘ 𝐺 )
2		lsmcntz.s	⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
3		lsmcntz.t	⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
4		lsmcntz.u	⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
5		lsmdisj.o	⊢ 0 = ( 0g ‘ 𝐺 )
6		lsmdisj.i	⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } )
7	1	lsmub1	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑆 ⊆ ( 𝑆 ⊕ 𝑇 ) )
8	2 3 7	syl2anc	⊢ ( 𝜑 → 𝑆 ⊆ ( 𝑆 ⊕ 𝑇 ) )
9	8	ssrind	⊢ ( 𝜑 → ( 𝑆 ∩ 𝑈 ) ⊆ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) )
10	9 6	sseqtrd	⊢ ( 𝜑 → ( 𝑆 ∩ 𝑈 ) ⊆ { 0 } )
11	5	subg0cl	⊢ ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ 𝑆 )
12	2 11	syl	⊢ ( 𝜑 → 0 ∈ 𝑆 )
13	5	subg0cl	⊢ ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ 𝑈 )
14	4 13	syl	⊢ ( 𝜑 → 0 ∈ 𝑈 )
15	12 14	elind	⊢ ( 𝜑 → 0 ∈ ( 𝑆 ∩ 𝑈 ) )
16	15	snssd	⊢ ( 𝜑 → { 0 } ⊆ ( 𝑆 ∩ 𝑈 ) )
17	10 16	eqssd	⊢ ( 𝜑 → ( 𝑆 ∩ 𝑈 ) = { 0 } )
18	1	lsmub2	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑇 ⊆ ( 𝑆 ⊕ 𝑇 ) )
19	2 3 18	syl2anc	⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑆 ⊕ 𝑇 ) )
20	19	ssrind	⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) ⊆ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) )
21	20 6	sseqtrd	⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) ⊆ { 0 } )
22	5	subg0cl	⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 0 ∈ 𝑇 )
23	3 22	syl	⊢ ( 𝜑 → 0 ∈ 𝑇 )
24	23 14	elind	⊢ ( 𝜑 → 0 ∈ ( 𝑇 ∩ 𝑈 ) )
25	24	snssd	⊢ ( 𝜑 → { 0 } ⊆ ( 𝑇 ∩ 𝑈 ) )
26	21 25	eqssd	⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } )
27	17 26	jca	⊢ ( 𝜑 → ( ( 𝑆 ∩ 𝑈 ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) )