Description: The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | ipffval.1 | ⊢ 𝑉 = ( Base ‘ 𝑊 ) | |
ipffval.2 | ⊢ , = ( ·𝑖 ‘ 𝑊 ) | ||
ipffval.3 | ⊢ · = ( ·if ‘ 𝑊 ) | ||
Assertion | ipfval | ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 , 𝑌 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ipffval.1 | ⊢ 𝑉 = ( Base ‘ 𝑊 ) | |
2 | ipffval.2 | ⊢ , = ( ·𝑖 ‘ 𝑊 ) | |
3 | ipffval.3 | ⊢ · = ( ·if ‘ 𝑊 ) | |
4 | oveq12 | ⊢ ( ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) → ( 𝑥 , 𝑦 ) = ( 𝑋 , 𝑌 ) ) | |
5 | 1 2 3 | ipffval | ⊢ · = ( 𝑥 ∈ 𝑉 , 𝑦 ∈ 𝑉 ↦ ( 𝑥 , 𝑦 ) ) |
6 | ovex | ⊢ ( 𝑋 , 𝑌 ) ∈ V | |
7 | 4 5 6 | ovmpoa | ⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 · 𝑌 ) = ( 𝑋 , 𝑌 ) ) |