This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 } ) ) )