Metamath Proof Explorer

Theorem lsmdisj2a

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 )
Assertion lsmdisj2a ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 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 2 adantr ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
7 3 adantr ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
8 4 adantr ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
9 simprl ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } )
10 simprr ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑇 ) = { 0 } )
11 1 6 7 8 5 9 10 lsmdisj2 ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } )
12 1 6 7 8 5 9 lsmdisj ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( ( 𝑆 ∩ 𝑈 ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) )
13 12 simpld ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑈 ) = { 0 } )
14 11 13 jca ⊢ ( ( 𝜑 ∧ ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ) → ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) )
15 incom ⊢ ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = ( 𝑈 ∩ ( 𝑆 ⊕ 𝑇 ) )
16 2 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
17 4 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
18 3 adantr ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
19 incom ⊢ ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) )
20 simprl ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } )
21 19 20 syl5eq ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( 𝑆 ⊕ 𝑈 ) ∩ 𝑇 ) = { 0 } )
22 simprr ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑈 ) = { 0 } )
23 1 16 17 18 5 21 22 lsmdisj2 ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑈 ∩ ( 𝑆 ⊕ 𝑇 ) ) = { 0 } )
24 15 23 syl5eq ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } )
25 incom ⊢ ( 𝑆 ∩ 𝑇 ) = ( 𝑇 ∩ 𝑆 )
26 1 18 16 17 5 20 lsmdisjr ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( 𝑇 ∩ 𝑆 ) = { 0 } ∧ ( 𝑇 ∩ 𝑈 ) = { 0 } ) )
27 26 simpld ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑇 ∩ 𝑆 ) = { 0 } )
28 25 27 syl5eq ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( 𝑆 ∩ 𝑇 ) = { 0 } )
29 24 28 jca ⊢ ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) → ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) )
30 14 29 impbida ⊢ ( 𝜑 → ( ( ( ( 𝑆 ⊕ 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆 ∩ 𝑇 ) = { 0 } ) ↔ ( ( 𝑇 ∩ ( 𝑆 ⊕ 𝑈 ) ) = { 0 } ∧ ( 𝑆 ∩ 𝑈 ) = { 0 } ) ) )