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Metamath Proof Explorer
Description: The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015)
|
|
Ref |
Expression |
|
Hypothesis |
nlmnrg.1 |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
|
Assertion |
nlmngp2 |
⊢ ( 𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nlmnrg.1 |
⊢ 𝐹 = ( Scalar ‘ 𝑊 ) |
| 2 |
1
|
nlmnrg |
⊢ ( 𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing ) |
| 3 |
|
nrgngp |
⊢ ( 𝐹 ∈ NrmRing → 𝐹 ∈ NrmGrp ) |
| 4 |
2 3
|
syl |
⊢ ( 𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp ) |