Step |
Hyp |
Ref |
Expression |
1 |
|
lcvexch.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
2 |
|
lcvexch.p |
⊢ ⊕ = ( LSSum ‘ 𝑊 ) |
3 |
|
lcvexch.c |
⊢ 𝐶 = ( ⋖L ‘ 𝑊 ) |
4 |
|
lcvexch.w |
⊢ ( 𝜑 → 𝑊 ∈ LMod ) |
5 |
|
lcvexch.t |
⊢ ( 𝜑 → 𝑇 ∈ 𝑆 ) |
6 |
|
lcvexch.u |
⊢ ( 𝜑 → 𝑈 ∈ 𝑆 ) |
7 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑊 ∈ LMod ) |
8 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑇 ∈ 𝑆 ) |
9 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑈 ∈ 𝑆 ) |
10 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) |
11 |
1 2 3 7 8 9 10
|
lcvexchlem5 |
⊢ ( ( 𝜑 ∧ ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) → 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) |
12 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑊 ∈ LMod ) |
13 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑇 ∈ 𝑆 ) |
14 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑈 ∈ 𝑆 ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) |
16 |
1 2 3 12 13 14 15
|
lcvexchlem4 |
⊢ ( ( 𝜑 ∧ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) → ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ) |
17 |
11 16
|
impbida |
⊢ ( 𝜑 → ( ( 𝑇 ∩ 𝑈 ) 𝐶 𝑈 ↔ 𝑇 𝐶 ( 𝑇 ⊕ 𝑈 ) ) ) |