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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 20613), while ·𝑖 only has closure (ipcl 20595). (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 20587 | . 2 class ·if | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3408 | . . 3 class V | |
4 | vx | . . . 4 setvar 𝑥 | |
5 | vy | . . . 4 setvar 𝑦 | |
6 | 2 | cv 1542 | . . . . 5 class 𝑔 |
7 | cbs 16760 | . . . . 5 class Base | |
8 | 6, 7 | cfv 6380 | . . . 4 class (Base‘𝑔) |
9 | 4 | cv 1542 | . . . . 5 class 𝑥 |
10 | 5 | cv 1542 | . . . . 5 class 𝑦 |
11 | cip 16807 | . . . . . 6 class ·𝑖 | |
12 | 6, 11 | cfv 6380 | . . . . 5 class (·𝑖‘𝑔) |
13 | 9, 10, 12 | co 7213 | . . . 4 class (𝑥(·𝑖‘𝑔)𝑦) |
14 | 4, 5, 8, 8, 13 | cmpo 7215 | . . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) |
15 | 2, 3, 14 | cmpt 5135 | . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
16 | 1, 15 | wceq 1543 | 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 20610 |
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