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Metamath Proof Explorer

Theorem lsmdisj3b

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 )
lsmdisj3b.2 ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) )
Assertion lsmdisj3b ( 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 lsmdisj3b.2 ⊢ ( 𝜑 → 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) )
8 1 2 4 3 5 lsmdisj2b ⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) = { 0 } ∧ ( 𝑈 ∩ 𝑇 ) = { 0 } ) ) )
9 1 6 lsmcom2 ⊢ ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ⊆ ( 𝑍 ‘ 𝑈 ) ) → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) )
10 3 4 7 9 syl3anc ⊢ ( 𝜑 → ( 𝑇 ⊕ 𝑈 ) = ( 𝑈 ⊕ 𝑇 ) )
11 10 ineq2d ⊢ ( 𝜑 → ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) )
12 11 eqeq1d ⊢ ( 𝜑 → ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ↔ ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) = { 0 } ) )
13 incom ⊢ ( 𝑇 ∩ 𝑈 ) = ( 𝑈 ∩ 𝑇 )
14 13 a1i ⊢ ( 𝜑 → ( 𝑇 ∩ 𝑈 ) = ( 𝑈 ∩ 𝑇 ) )
15 14 eqeq1d ⊢ ( 𝜑 → ( ( 𝑇 ∩ 𝑈 ) = { 0 } ↔ ( 𝑈 ∩ 𝑇 ) = { 0 } ) )
16 12 15 anbi12d ⊢ ( 𝜑 → ( ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑈 ⊕ 𝑇 ) ) = { 0 } ∧ ( 𝑈 ∩ 𝑇 ) = { 0 } ) ) )
17 8 16 bitr4d ⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑆 ∩ ( 𝑇 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) ) )