Metamath Proof Explorer
Description: Value of the prime-counting function pi for 1, according to Ján
Mináč. (Contributed by AV, 4-Apr-2026)
|
|
Ref |
Expression |
|
Assertion |
ppi1sum |
⊢ ( π ‘ 1 ) = Σ 𝑘 ∈ ∅ ( ⌊ ‘ ( ( ( ( ! ‘ ( 𝑘 − 1 ) ) + 1 ) / 𝑘 ) − ( ⌊ ‘ ( ( ! ‘ ( 𝑘 − 1 ) ) / 𝑘 ) ) ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
ppi1 |
⊢ ( π ‘ 1 ) = 0 |
| 2 |
|
sum0 |
⊢ Σ 𝑘 ∈ ∅ ( ⌊ ‘ ( ( ( ( ! ‘ ( 𝑘 − 1 ) ) + 1 ) / 𝑘 ) − ( ⌊ ‘ ( ( ! ‘ ( 𝑘 − 1 ) ) / 𝑘 ) ) ) ) = 0 |
| 3 |
1 2
|
eqtr4i |
⊢ ( π ‘ 1 ) = Σ 𝑘 ∈ ∅ ( ⌊ ‘ ( ( ( ( ! ‘ ( 𝑘 − 1 ) ) + 1 ) / 𝑘 ) − ( ⌊ ‘ ( ( ! ‘ ( 𝑘 − 1 ) ) / 𝑘 ) ) ) ) |