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 20723), while ·𝑖 only has closure (ipcl 20705). (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 20697 | . 2 class ·if | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3492 | . . 3 class V | |
4 | vx | . . . 4 setvar 𝑥 | |
5 | vy | . . . 4 setvar 𝑦 | |
6 | 2 | cv 1527 | . . . . 5 class 𝑔 |
7 | cbs 16471 | . . . . 5 class Base | |
8 | 6, 7 | cfv 6348 | . . . 4 class (Base‘𝑔) |
9 | 4 | cv 1527 | . . . . 5 class 𝑥 |
10 | 5 | cv 1527 | . . . . 5 class 𝑦 |
11 | cip 16558 | . . . . . 6 class ·𝑖 | |
12 | 6, 11 | cfv 6348 | . . . . 5 class (·𝑖‘𝑔) |
13 | 9, 10, 12 | co 7145 | . . . 4 class (𝑥(·𝑖‘𝑔)𝑦) |
14 | 4, 5, 8, 8, 13 | cmpo 7147 | . . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) |
15 | 2, 3, 14 | cmpt 5137 | . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
16 | 1, 15 | wceq 1528 | 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 20720 |
Copyright terms: Public domain | W3C validator |