Step |
Hyp |
Ref |
Expression |
1 |
|
lsmless2.v |
⊢ 𝐵 = ( Base ‘ 𝐺 ) |
2 |
|
lsmless2.s |
⊢ ⊕ = ( LSSum ‘ 𝐺 ) |
3 |
|
ssrexv |
⊢ ( 𝑅 ⊆ 𝑇 → ( ∃ 𝑦 ∈ 𝑅 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
4 |
3
|
adantl |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → ( ∃ 𝑦 ∈ 𝑅 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) → ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
5 |
|
simpl1 |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → 𝐺 ∈ 𝑉 ) |
6 |
|
simpr |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → 𝑅 ⊆ 𝑇 ) |
7 |
|
simpl2 |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → 𝑇 ⊆ 𝐵 ) |
8 |
6 7
|
sstrd |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → 𝑅 ⊆ 𝐵 ) |
9 |
|
simpl3 |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → 𝑈 ⊆ 𝐵 ) |
10 |
|
eqid |
⊢ ( +g ‘ 𝐺 ) = ( +g ‘ 𝐺 ) |
11 |
1 10 2
|
lsmelvalx |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑅 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑥 ∈ ( 𝑅 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑅 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
12 |
5 8 9 11
|
syl3anc |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → ( 𝑥 ∈ ( 𝑅 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑅 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
13 |
1 10 2
|
lsmelvalx |
⊢ ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) → ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
14 |
13
|
adantr |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → ( 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ↔ ∃ 𝑦 ∈ 𝑇 ∃ 𝑧 ∈ 𝑈 𝑥 = ( 𝑦 ( +g ‘ 𝐺 ) 𝑧 ) ) ) |
15 |
4 12 14
|
3imtr4d |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → ( 𝑥 ∈ ( 𝑅 ⊕ 𝑈 ) → 𝑥 ∈ ( 𝑇 ⊕ 𝑈 ) ) ) |
16 |
15
|
ssrdv |
⊢ ( ( ( 𝐺 ∈ 𝑉 ∧ 𝑇 ⊆ 𝐵 ∧ 𝑈 ⊆ 𝐵 ) ∧ 𝑅 ⊆ 𝑇 ) → ( 𝑅 ⊕ 𝑈 ) ⊆ ( 𝑇 ⊕ 𝑈 ) ) |