Metamath Proof Explorer


Theorem lsmss1

Description: Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014) (Revised by Mario Carneiro, 19-Apr-2016)

Ref Expression
Hypothesis lsmub1.p = ( LSSum ‘ 𝐺 )
Assertion lsmss1 ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇𝑈 ) → ( 𝑇 𝑈 ) = 𝑈 )

Proof

Step Hyp Ref Expression
1 lsmub1.p = ( LSSum ‘ 𝐺 )
2 ssid 𝑈𝑈
3 1 lsmlub ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑇𝑈𝑈𝑈 ) ↔ ( 𝑇 𝑈 ) ⊆ 𝑈 ) )
4 3 3anidm23 ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑇𝑈𝑈𝑈 ) ↔ ( 𝑇 𝑈 ) ⊆ 𝑈 ) )
5 4 biimpd ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( ( 𝑇𝑈𝑈𝑈 ) → ( 𝑇 𝑈 ) ⊆ 𝑈 ) )
6 2 5 mpan2i ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → ( 𝑇𝑈 → ( 𝑇 𝑈 ) ⊆ 𝑈 ) )
7 6 3impia ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇𝑈 ) → ( 𝑇 𝑈 ) ⊆ 𝑈 )
8 1 lsmub2 ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ) → 𝑈 ⊆ ( 𝑇 𝑈 ) )
9 8 3adant3 ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇𝑈 ) → 𝑈 ⊆ ( 𝑇 𝑈 ) )
10 7 9 eqssd ( ( 𝑇 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑈 ∈ ( SubGrp ‘ 𝐺 ) ∧ 𝑇𝑈 ) → ( 𝑇 𝑈 ) = 𝑈 )