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 ( 𝜑𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
lsmcntz.t ( 𝜑𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
lsmcntz.u ( 𝜑𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
lsmdisj.o 0 = ( 0g𝐺 )
Assertion lsmdisj2a ( 𝜑 → ( ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ↔ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) )

Proof

Step Hyp Ref Expression
1 lsmcntz.p = ( LSSum ‘ 𝐺 )
2 lsmcntz.s ( 𝜑𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
3 lsmcntz.t ( 𝜑𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
4 lsmcntz.u ( 𝜑𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
5 lsmdisj.o 0 = ( 0g𝐺 )
6 2 adantr ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
7 3 adantr ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
8 4 adantr ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
9 simprl ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } )
10 simprr ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → ( 𝑆𝑇 ) = { 0 } )
11 1 6 7 8 5 9 10 lsmdisj2 ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } )
12 1 6 7 8 5 9 lsmdisj ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → ( ( 𝑆𝑈 ) = { 0 } ∧ ( 𝑇𝑈 ) = { 0 } ) )
13 12 simpld ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → ( 𝑆𝑈 ) = { 0 } )
14 11 13 jca ( ( 𝜑 ∧ ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ) → ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) )
15 incom ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = ( 𝑈 ∩ ( 𝑆 𝑇 ) )
16 2 adantr ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → 𝑆 ∈ ( SubGrp ‘ 𝐺 ) )
17 4 adantr ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → 𝑈 ∈ ( SubGrp ‘ 𝐺 ) )
18 3 adantr ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → 𝑇 ∈ ( SubGrp ‘ 𝐺 ) )
19 incom ( ( 𝑆 𝑈 ) ∩ 𝑇 ) = ( 𝑇 ∩ ( 𝑆 𝑈 ) )
20 simprl ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } )
21 19 20 syl5eq ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( ( 𝑆 𝑈 ) ∩ 𝑇 ) = { 0 } )
22 simprr ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( 𝑆𝑈 ) = { 0 } )
23 1 16 17 18 5 21 22 lsmdisj2 ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( 𝑈 ∩ ( 𝑆 𝑇 ) ) = { 0 } )
24 15 23 syl5eq ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } )
25 incom ( 𝑆𝑇 ) = ( 𝑇𝑆 )
26 1 18 16 17 5 20 lsmdisjr ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( ( 𝑇𝑆 ) = { 0 } ∧ ( 𝑇𝑈 ) = { 0 } ) )
27 26 simpld ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( 𝑇𝑆 ) = { 0 } )
28 25 27 syl5eq ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( 𝑆𝑇 ) = { 0 } )
29 24 28 jca ( ( 𝜑 ∧ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) → ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) )
30 14 29 impbida ( 𝜑 → ( ( ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } ∧ ( 𝑆𝑇 ) = { 0 } ) ↔ ( ( 𝑇 ∩ ( 𝑆 𝑈 ) ) = { 0 } ∧ ( 𝑆𝑈 ) = { 0 } ) ) )