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Metamath Proof Explorer
Description: A subcomplex vector space is a subcomplex module. (Contributed by Thierry Arnoux, 22-May-2019)
|
|
Ref |
Expression |
|
Hypothesis |
cvslvec.1 |
⊢ ( 𝜑 → 𝑊 ∈ ℂVec ) |
|
Assertion |
cvsclm |
⊢ ( 𝜑 → 𝑊 ∈ ℂMod ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
cvslvec.1 |
⊢ ( 𝜑 → 𝑊 ∈ ℂVec ) |
| 2 |
|
df-cvs |
⊢ ℂVec = ( ℂMod ∩ LVec ) |
| 3 |
2
|
elin2 |
⊢ ( 𝑊 ∈ ℂVec ↔ ( 𝑊 ∈ ℂMod ∧ 𝑊 ∈ LVec ) ) |
| 4 |
3
|
simplbi |
⊢ ( 𝑊 ∈ ℂVec → 𝑊 ∈ ℂMod ) |
| 5 |
1 4
|
syl |
⊢ ( 𝜑 → 𝑊 ∈ ℂMod ) |