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Metamath Proof Explorer
Description: The scalar component of a left module is an additive group.
(Contributed by NM, 8-Dec-2013) (Revised by Mario Carneiro, 19-Jun-2014)
|
|
Ref |
Expression |
|
Hypothesis |
lmodring.1 |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
|
Assertion |
lmodfgrp |
⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Grp ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
lmodring.1 |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
| 2 |
1
|
lmodring |
⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Ring ) |
| 3 |
|
ringgrp |
⊢ ( 𝐹 ∈ Ring → 𝐹 ∈ Grp ) |
| 4 |
2 3
|
syl |
⊢ ( 𝑊 ∈ LMod → 𝐹 ∈ Grp ) |