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Metamath Proof Explorer

Definition df-ipf

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 ), while .i only has closure ( ipcl ). (Contributed by Mario Carneiro, 12-Aug-2015)

Ref Expression
Assertion df-ipf ·if = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( ·𝑖 ‘ 𝑔 ) 𝑦 ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cipf ⊢ ·if
1 vg ⊢ 𝑔
2 cvv ⊢ V
3 vx ⊢ 𝑥
4 cbs ⊢ Base
5 1 cv ⊢ 𝑔
6 5 4 cfv ⊢ ( Base ‘ 𝑔 )
7 vy ⊢ 𝑦
8 3 cv ⊢ 𝑥
9 cip ⊢ ·𝑖
10 5 9 cfv ⊢ ( ·𝑖 ‘ 𝑔 )
11 7 cv ⊢ 𝑦
12 8 11 10 co ⊢ ( 𝑥 ( ·𝑖 ‘ 𝑔 ) 𝑦 )
13 3 7 6 6 12 cmpo ⊢ ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( ·𝑖 ‘ 𝑔 ) 𝑦 ) )
14 1 2 13 cmpt ⊢ ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( ·𝑖 ‘ 𝑔 ) 𝑦 ) ) )
15 0 14 wceq ⊢ ·if = ( 𝑔 ∈ V ↦ ( 𝑥 ∈ ( Base ‘ 𝑔 ) , 𝑦 ∈ ( Base ‘ 𝑔 ) ↦ ( 𝑥 ( ·𝑖 ‘ 𝑔 ) 𝑦 ) ) )