| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > df-ipf | Structured version Visualization version GIF version | ||
| Description: Define the inner product function. Usually we will use ·𝑖 directly instead of ·if, and they have the same behavior in most cases. The main advantage of ·if is that it is a guaranteed function (ipffn 21669), while ·𝑖 only has closure (ipcl 21651). (Contributed by Mario Carneiro, 12-Aug-2015.) |
| Ref | Expression |
|---|---|
| df-ipf | ⊢ ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cipf 21643 | . 2 class ·if | |
| 2 | vg | . . 3 setvar 𝑔 | |
| 3 | cvv 3480 | . . 3 class V | |
| 4 | vx | . . . 4 setvar 𝑥 | |
| 5 | vy | . . . 4 setvar 𝑦 | |
| 6 | 2 | cv 1539 | . . . . 5 class 𝑔 |
| 7 | cbs 17247 | . . . . 5 class Base | |
| 8 | 6, 7 | cfv 6561 | . . . 4 class (Base‘𝑔) |
| 9 | 4 | cv 1539 | . . . . 5 class 𝑥 |
| 10 | 5 | cv 1539 | . . . . 5 class 𝑦 |
| 11 | cip 17302 | . . . . . 6 class ·𝑖 | |
| 12 | 6, 11 | cfv 6561 | . . . . 5 class (·𝑖‘𝑔) |
| 13 | 9, 10, 12 | co 7431 | . . . 4 class (𝑥(·𝑖‘𝑔)𝑦) |
| 14 | 4, 5, 8, 8, 13 | cmpo 7433 | . . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) |
| 15 | 2, 3, 14 | cmpt 5225 | . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
| 16 | 1, 15 | wceq 1540 | 1 wff ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
| Colors of variables: wff setvar class |
| This definition is referenced by: ipffval 21666 |
| Copyright terms: Public domain | W3C validator |