Metamath Proof Explorer
Description: The inner product is unaffected by restriction. (Contributed by Thierry Arnoux, 16-Jun-2019)
|
|
Ref |
Expression |
|
Hypotheses |
resssca.1 |
⊢ 𝐻 = ( 𝐺 ↾s 𝐴 ) |
|
|
ressip.2 |
⊢ , = ( ·𝑖 ‘ 𝐺 ) |
|
Assertion |
ressip |
⊢ ( 𝐴 ∈ 𝑉 → , = ( ·𝑖 ‘ 𝐻 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
resssca.1 |
⊢ 𝐻 = ( 𝐺 ↾s 𝐴 ) |
2 |
|
ressip.2 |
⊢ , = ( ·𝑖 ‘ 𝐺 ) |
3 |
|
df-ip |
⊢ ·𝑖 = Slot 8 |
4 |
|
8nn |
⊢ 8 ∈ ℕ |
5 |
|
1lt8 |
⊢ 1 < 8 |
6 |
1 2 3 4 5
|
resslem |
⊢ ( 𝐴 ∈ 𝑉 → , = ( ·𝑖 ‘ 𝐻 ) ) |