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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 20340), while ·𝑖 only has closure (ipcl 20322). (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 20314 | . 2 class ·if | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3441 | . . 3 class V | |
4 | vx | . . . 4 setvar 𝑥 | |
5 | vy | . . . 4 setvar 𝑦 | |
6 | 2 | cv 1537 | . . . . 5 class 𝑔 |
7 | cbs 16475 | . . . . 5 class Base | |
8 | 6, 7 | cfv 6324 | . . . 4 class (Base‘𝑔) |
9 | 4 | cv 1537 | . . . . 5 class 𝑥 |
10 | 5 | cv 1537 | . . . . 5 class 𝑦 |
11 | cip 16562 | . . . . . 6 class ·𝑖 | |
12 | 6, 11 | cfv 6324 | . . . . 5 class (·𝑖‘𝑔) |
13 | 9, 10, 12 | co 7135 | . . . 4 class (𝑥(·𝑖‘𝑔)𝑦) |
14 | 4, 5, 8, 8, 13 | cmpo 7137 | . . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦)) |
15 | 2, 3, 14 | cmpt 5110 | . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖‘𝑔)𝑦))) |
16 | 1, 15 | wceq 1538 | 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 20337 |
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