Theorem lsmdisj3a

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 )
	lsmdisj3b.z	⊢ 𝑍 = ( Cntz ‘ 𝐺 )
	lsmdisj3a.2	⊢ ( 𝜑 → 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) )
Assertion	lsmdisj3a	⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑇 ⊕ 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		lsmdisj3b.z	⊢ 𝑍 = ( Cntz ‘ 𝐺 )
7		lsmdisj3a.2	⊢ ( 𝜑 → 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) )
8	1 6	lsmcom2	⊢ ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑆 ⊆ ( 𝑍 ‘ 𝑇 ) ) → ( 𝑆 ⊕ 𝑇 ) = ( 𝑇 ⊕ 𝑆 ) )
9	2 3 7 8	syl3anc	⊢ ( 𝜑 → ( 𝑆 ⊕ 𝑇 ) = ( 𝑇 ⊕ 𝑆 ) )
10	9	ineq1d	⊢ ( 𝜑 → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) )
11	10	eqeq1d	⊢ ( 𝜑 → ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ↔ ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) = { 0 } ) )
12		incom	⊢ ( 𝑆 ∩ 𝑇 ) = ( 𝑇 ∩ 𝑆 )
13	12	a1i	⊢ ( 𝜑 → ( 𝑆 ∩ 𝑇 ) = ( 𝑇 ∩ 𝑆 ) )
14	13	eqeq1d	⊢ ( 𝜑 → ( ( 𝑆 ∩ 𝑇 ) = { 0 } ↔ ( 𝑇 ∩ 𝑆 ) = { 0 } ) )
15	11 14	anbi12d	⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑇 ∩ 𝑆 ) = { 0 } ) ) )
16	1 3 2 4 5	lsmdisj2a	⊢ ( 𝜑 → ( ( ( ( 𝑇 ⊕ 𝑆 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑇 ∩ 𝑆 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ) )
17	15 16	bitrd	⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ) )