This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Difference of a Hilbert space operator from itself. (Contributed by NM, 10-Mar-2006) (New usage is discouraged.)
|
|
Ref |
Expression |
|
Hypothesis |
hoaddrid.1 |
⊢ 𝑇 : ℋ ⟶ ℋ |
|
Assertion |
hodidi |
⊢ ( 𝑇 −op 𝑇 ) = 0hop |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hoaddrid.1 |
⊢ 𝑇 : ℋ ⟶ ℋ |
| 2 |
1
|
hoaddridi |
⊢ ( 𝑇 +op 0hop ) = 𝑇 |
| 3 |
|
ho0f |
⊢ 0hop : ℋ ⟶ ℋ |
| 4 |
1 1 3
|
hodsi |
⊢ ( ( 𝑇 −op 𝑇 ) = 0hop ↔ ( 𝑇 +op 0hop ) = 𝑇 ) |
| 5 |
2 4
|
mpbir |
⊢ ( 𝑇 −op 𝑇 ) = 0hop |