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Metamath Proof Explorer

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 } )