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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 19996), while ·𝑖 only has closure (ipcl 19978). (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 19970 | . 2 class ·if | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3200 | . . 3 class V | |
4 | vx | . . . 4 setvar 𝑥 | |
5 | vy | . . . 4 setvar 𝑦 | |
6 | 2 | cv 1482 | . . . . 5 class 𝑔 |
7 | cbs 15857 | . . . . 5 class Base | |
8 | 6, 7 | cfv 5888 | . . . 4 class (Base‘𝑔) |
9 | 4 | cv 1482 | . . . . 5 class 𝑥 |
10 | 5 | cv 1482 | . . . . 5 class 𝑦 |
11 | cip 15946 | . . . . . 6 class ·𝑖 | |
12 | 6, 11 | cfv 5888 | . . . . 5 class (·𝑖‘𝑔) |
13 | 9, 10, 12 | co 6650 | . . . 4 class (𝑥(·𝑖‘𝑔)𝑦) |
14 | 4, 5, 8, 8, 13 | cmpt2 6652 | . . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) |
15 | 2, 3, 14 | cmpt 4729 | . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
16 | 1, 15 | wceq 1483 | 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 19993 |
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