Metamath Proof Explorer


Theorem lsmless1x

Description: Subset implies subgroup sum subset (extended domain version). (Contributed by NM, 22-Feb-2014) (Revised by Mario Carneiro, 19-Apr-2016)

Ref Expression
Hypotheses lsmless2.v 𝐵 = ( Base ‘ 𝐺 )
lsmless2.s = ( LSSum ‘ 𝐺 )
Assertion lsmless1x ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → ( 𝑅 𝑈 ) ⊆ ( 𝑇 𝑈 ) )

Proof

Step Hyp Ref Expression
1 lsmless2.v 𝐵 = ( Base ‘ 𝐺 )
2 lsmless2.s = ( LSSum ‘ 𝐺 )
3 ssrexv ( 𝑅𝑇 → ( ∃ 𝑦𝑅𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) → ∃ 𝑦𝑇𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) ) )
4 3 adantl ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → ( ∃ 𝑦𝑅𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) → ∃ 𝑦𝑇𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) ) )
5 simpl1 ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → 𝐺𝑉 )
6 simpr ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → 𝑅𝑇 )
7 simpl2 ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → 𝑇𝐵 )
8 6 7 sstrd ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → 𝑅𝐵 )
9 simpl3 ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → 𝑈𝐵 )
10 eqid ( +g𝐺 ) = ( +g𝐺 )
11 1 10 2 lsmelvalx ( ( 𝐺𝑉𝑅𝐵𝑈𝐵 ) → ( 𝑥 ∈ ( 𝑅 𝑈 ) ↔ ∃ 𝑦𝑅𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) ) )
12 5 8 9 11 syl3anc ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → ( 𝑥 ∈ ( 𝑅 𝑈 ) ↔ ∃ 𝑦𝑅𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) ) )
13 1 10 2 lsmelvalx ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) → ( 𝑥 ∈ ( 𝑇 𝑈 ) ↔ ∃ 𝑦𝑇𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) ) )
14 13 adantr ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → ( 𝑥 ∈ ( 𝑇 𝑈 ) ↔ ∃ 𝑦𝑇𝑧𝑈 𝑥 = ( 𝑦 ( +g𝐺 ) 𝑧 ) ) )
15 4 12 14 3imtr4d ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → ( 𝑥 ∈ ( 𝑅 𝑈 ) → 𝑥 ∈ ( 𝑇 𝑈 ) ) )
16 15 ssrdv ( ( ( 𝐺𝑉𝑇𝐵𝑈𝐵 ) ∧ 𝑅𝑇 ) → ( 𝑅 𝑈 ) ⊆ ( 𝑇 𝑈 ) )