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Definition df-ipf 21663
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.)
Assertion
Ref Expression
df-ipf ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
Distinct variable group:   𝑥,𝑔,𝑦

Detailed syntax breakdown of Definition df-ipf
StepHypRef 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
62cv 1536 . . . . 5 class 𝑔
7 cbs 17245 . . . . 5 class Base
86, 7cfv 6563 . . . 4 class (Base‘𝑔)
94cv 1536 . . . . 5 class 𝑥
105cv 1536 . . . . 5 class 𝑦
11 cip 17303 . . . . . 6 class ·𝑖
126, 11cfv 6563 . . . . 5 class (·𝑖𝑔)
139, 10, 12co 7431 . . . 4 class (𝑥(·𝑖𝑔)𝑦)
144, 5, 8, 8, 13cmpo 7433 . . 3 class (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦))
152, 3, 14cmpt 5231 . 2 class (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
161, 15wceq 1537 1 wff ·if = (𝑔 ∈ V ↦ (𝑥 ∈ (Base‘𝑔), 𝑦 ∈ (Base‘𝑔) ↦ (𝑥(·𝑖𝑔)𝑦)))
Colors of variables: wff setvar class
This definition is referenced by:  ipffval  21684
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