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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 21687), while ·𝑖 only has closure (ipcl 21669). (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 21661 | . 2 class ·if | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3478 | . . 3 class V | |
4 | vx | . . . 4 setvar 𝑥 | |
5 | vy | . . . 4 setvar 𝑦 | |
6 | 2 | cv 1536 | . . . . 5 class 𝑔 |
7 | cbs 17245 | . . . . 5 class Base | |
8 | 6, 7 | cfv 6563 | . . . 4 class (Base‘𝑔) |
9 | 4 | cv 1536 | . . . . 5 class 𝑥 |
10 | 5 | cv 1536 | . . . . 5 class 𝑦 |
11 | cip 17303 | . . . . . 6 class ·𝑖 | |
12 | 6, 11 | cfv 6563 | . . . . 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 5231 | . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
16 | 1, 15 | wceq 1537 | 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 21684 |
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