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Metamath Proof Explorer
Description: A linear operator with a nonzero norm is nonzero. (Contributed by NM, 12-Aug-2006) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Assertion |
nmlnopne0 |
⊢ ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) ≠ 0 ↔ 𝑇 ≠ 0hop ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
nmlnop0 |
⊢ ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) = 0 ↔ 𝑇 = 0hop ) ) |
| 2 |
1
|
necon3bid |
⊢ ( 𝑇 ∈ LinOp → ( ( normop ‘ 𝑇 ) ≠ 0 ↔ 𝑇 ≠ 0hop ) ) |