Theorem lsmdisjr

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 )
	lsmdisjr.i	⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } )
Assertion	lsmdisjr	⊢ ( 𝜑 → ( ( 𝑆 ∩ 𝑇 ) = { 0 } ∧ ( 𝑆 ∩ 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		lsmdisjr.i	⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } )
7		incom	⊢ ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 )
8	7 6	eqtr3id	⊢ ( 𝜑 → ( ( 𝑇 ⊕ 𝑈 ) ∩ 𝑆 ) = { 0 } )
9	1 3 4 2 5 8	lsmdisj	⊢ ( 𝜑 → ( ( 𝑇 ∩ 𝑆 ) = { 0 } ∧ ( 𝑈 ∩ 𝑆 ) = { 0 } ) )
10		incom	⊢ ( 𝑇 ∩ 𝑆 ) = ( 𝑆 ∩ 𝑇 )
11	10	eqeq1i	⊢ ( ( 𝑇 ∩ 𝑆 ) = { 0 } ↔ ( 𝑆 ∩ 𝑇 ) = { 0 } )
12		incom	⊢ ( 𝑈 ∩ 𝑆 ) = ( 𝑆 ∩ 𝑈 )
13	12	eqeq1i	⊢ ( ( 𝑈 ∩ 𝑆 ) = { 0 } ↔ ( 𝑆 ∩ 𝑈 ) = { 0 } )
14	11 13	anbi12i	⊢ ( ( ( 𝑇 ∩ 𝑆 ) = { 0 } ∧ ( 𝑈 ∩ 𝑆 ) = { 0 } ) ↔ ( ( 𝑆 ∩ 𝑇 ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) )
15	9 14	sylib	⊢ ( 𝜑 → ( ( 𝑆 ∩ 𝑇 ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) )