Theorem lsmdisj3

Description: Association of the disjointness constraint in 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 } )
	lsmdisj2.i	⊢ ( 𝜑 → ( 𝑆 ∩ 𝑇 ) = { 0 } )
	lsmdisj3.z	⊢ 𝑍 = ( Cntz ‘ 𝐺 )
	lsmdisj3.s	⊢ ( 𝜑 → 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) )
Assertion	lsmdisj3	⊢ ( 𝜑 → ( 𝑆 ∩ ( 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		lsmdisj2.i	⊢ ( 𝜑 → ( 𝑆 ∩ 𝑇 ) = { 0 } )
8		lsmdisj3.z	⊢ 𝑍 = ( Cntz ‘ 𝐺 )
9		lsmdisj3.s	⊢ ( 𝜑 → 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) )
10	1 8	lsmcom2	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ) → ( 𝑆 ⊕ 𝑇 ) = ( 𝑇 ⊕ 𝑆 ) )
11	2 3 9 10	syl3anc	⊢ ( 𝜑 → ( 𝑆 ⊕ 𝑇 ) = ( 𝑇 ⊕ 𝑆 ) )
12	11	ineq1d	⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) )
13	12 6	eqtr3d	⊢ ( 𝜑 → ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) = { 0 } )
14		incom	⊢ ( 𝑇 ∩ 𝑆 ) = ( 𝑆 ∩ 𝑇 )
15	14 7	eqtrid	⊢ ( 𝜑 → ( 𝑇 ∩ 𝑆 ) = { 0 } )
16	1 3 2 4 5 13 15	lsmdisj2	⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } )