Metamath Proof Explorer


Theorem lsmdisj

Description: Disjointness from 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 )
lsmdisj.i ( 𝜑 → ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } )
Assertion lsmdisj ( 𝜑 → ( ( 𝑆𝑈 ) = { 0 } ∧ ( 𝑇𝑈 ) = { 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 lsmdisj.i ( 𝜑 → ( ( 𝑆 𝑇 ) ∩ 𝑈 ) = { 0 } )
7 1 lsmub1 ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑆 ⊆ ( 𝑆 𝑇 ) )
8 2 3 7 syl2anc ( 𝜑𝑆 ⊆ ( 𝑆 𝑇 ) )
9 8 ssrind ( 𝜑 → ( 𝑆𝑈 ) ⊆ ( ( 𝑆 𝑇 ) ∩ 𝑈 ) )
10 9 6 sseqtrd ( 𝜑 → ( 𝑆𝑈 ) ⊆ { 0 } )
11 5 subg0cl ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) → 0𝑆 )
12 2 11 syl ( 𝜑0𝑆 )
13 5 subg0cl ( 𝑈 ∈ ( SubGrp ‘ 𝐺 ) → 0𝑈 )
14 4 13 syl ( 𝜑0𝑈 )
15 12 14 elind ( 𝜑0 ∈ ( 𝑆𝑈 ) )
16 15 snssd ( 𝜑 → { 0 } ⊆ ( 𝑆𝑈 ) )
17 10 16 eqssd ( 𝜑 → ( 𝑆𝑈 ) = { 0 } )
18 1 lsmub2 ( ( 𝑆 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑇 ⊆ ( 𝑆 𝑇 ) )
19 2 3 18 syl2anc ( 𝜑𝑇 ⊆ ( 𝑆 𝑇 ) )
20 19 ssrind ( 𝜑 → ( 𝑇𝑈 ) ⊆ ( ( 𝑆 𝑇 ) ∩ 𝑈 ) )
21 20 6 sseqtrd ( 𝜑 → ( 𝑇𝑈 ) ⊆ { 0 } )
22 5 subg0cl ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) → 0𝑇 )
23 3 22 syl ( 𝜑0𝑇 )
24 23 14 elind ( 𝜑0 ∈ ( 𝑇𝑈 ) )
25 24 snssd ( 𝜑 → { 0 } ⊆ ( 𝑇𝑈 ) )
26 21 25 eqssd ( 𝜑 → ( 𝑇𝑈 ) = { 0 } )
27 17 26 jca ( 𝜑 → ( ( 𝑆𝑈 ) = { 0 } ∧ ( 𝑇𝑈 ) = { 0 } ) )