Metamath Proof Explorer
Description: dist is unaffected by restriction. (Contributed by Mario Carneiro, 26-Aug-2015)
|
|
Ref |
Expression |
|
Hypotheses |
ressds.1 |
⊢ 𝐻 = ( 𝐺 ↾s 𝐴 ) |
|
|
ressds.2 |
⊢ 𝐷 = ( dist ‘ 𝐺 ) |
|
Assertion |
ressds |
⊢ ( 𝐴 ∈ 𝑉 → 𝐷 = ( dist ‘ 𝐻 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ressds.1 |
⊢ 𝐻 = ( 𝐺 ↾s 𝐴 ) |
2 |
|
ressds.2 |
⊢ 𝐷 = ( dist ‘ 𝐺 ) |
3 |
|
df-ds |
⊢ dist = Slot ; 1 2 |
4 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
5 |
|
2nn |
⊢ 2 ∈ ℕ |
6 |
4 5
|
decnncl |
⊢ ; 1 2 ∈ ℕ |
7 |
|
1nn |
⊢ 1 ∈ ℕ |
8 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
9 |
|
1lt10 |
⊢ 1 < ; 1 0 |
10 |
7 8 4 9
|
declti |
⊢ 1 < ; 1 2 |
11 |
1 2 3 6 10
|
resslem |
⊢ ( 𝐴 ∈ 𝑉 → 𝐷 = ( dist ‘ 𝐻 ) ) |