Theorem lsmdisj2r

Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-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 )
	lsmdisjr.i	⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } )
	lsmdisj2r.i	⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } )
Assertion	lsmdisj2r	⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = { 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		lsmdisjr.i	⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } )
7		lsmdisj2r.i	⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = { 0 } )
8		eqid	⊢ ( oppg ‘ 𝐺 ) = ( oppg ‘ 𝐺 )
9	8 1	oppglsm	⊢ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) = ( 𝑆 ⊕ 𝑈 )
10	9	ineq2i	⊢ ( 𝑇 ∩ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) ) = ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) )
11		incom	⊢ ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 )
12	10 11	eqtri	⊢ ( 𝑇 ∩ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) ) = ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 )
13		eqid	⊢ ( LSSum ‘ ( oppg ‘ 𝐺 ) ) = ( LSSum ‘ ( oppg ‘ 𝐺 ) )
14	8	oppgsubg	⊢ ( SubGrp ‘ 𝐺 ) = ( SubGrp ‘ ( oppg ‘ 𝐺 ) )
15	4 14	eleqtrdi	⊢ ( 𝜑 → 𝑈 ∈ ( SubGrp ‘ ( oppg ‘ 𝐺 ) ) )
16	3 14	eleqtrdi	⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ ( oppg ‘ 𝐺 ) ) )
17	2 14	eleqtrdi	⊢ ( 𝜑 → 𝑆 ∈ ( SubGrp ‘ ( oppg ‘ 𝐺 ) ) )
18	8 5	oppgid	⊢ 0 = ( 0g ‘ ( oppg ‘ 𝐺 ) )
19	8 1	oppglsm	⊢ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) = ( 𝑇 ⊕ 𝑈 )
20	19	ineq1i	⊢ ( ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) ∩ 𝑆 ) = ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 )
21		incom	⊢ ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 ) = ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) )
22	20 21	eqtri	⊢ ( ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) ∩ 𝑆 ) = ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) )
23	22 6	eqtrid	⊢ ( 𝜑 → ( ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑇 ) ∩ 𝑆 ) = { 0 } )
24		incom	⊢ ( 𝑇 ∩ 𝑈 ) = ( 𝑈 ∩ 𝑇 )
25	24 7	eqtr3id	⊢ ( 𝜑 → ( 𝑈 ∩ 𝑇 ) = { 0 } )
26	13 15 16 17 18 23 25	lsmdisj2	⊢ ( 𝜑 → ( 𝑇 ∩ ( 𝑈 ( LSSum ‘ ( oppg ‘ 𝐺 ) ) 𝑆 ) ) = { 0 } )
27	12 26	eqtr3id	⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = { 0 } )