Step |
Hyp |
Ref |
Expression |
1 |
|
hashrepr.a |
⊢ ( 𝜑 → 𝐴 ⊆ ℕ ) |
2 |
|
hashrepr.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
3 |
|
hashrepr.s |
⊢ ( 𝜑 → 𝑆 ∈ ℕ0 ) |
4 |
2
|
nn0zd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
5 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ∈ Fin ) |
6 |
|
fz1ssnn |
⊢ ( 1 ... 𝑀 ) ⊆ ℕ |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( 1 ... 𝑀 ) ⊆ ℕ ) |
8 |
1 4 3 5 7
|
hashreprin |
⊢ ( 𝜑 → ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑀 ) ) ( repr ‘ 𝑆 ) 𝑀 ) ) = Σ 𝑐 ∈ ( ( 1 ... 𝑀 ) ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |
9 |
2 3 1
|
reprinfz1 |
⊢ ( 𝜑 → ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) = ( ( 𝐴 ∩ ( 1 ... 𝑀 ) ) ( repr ‘ 𝑆 ) 𝑀 ) ) |
10 |
9
|
fveq2d |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ) = ( ♯ ‘ ( ( 𝐴 ∩ ( 1 ... 𝑀 ) ) ( repr ‘ 𝑆 ) 𝑀 ) ) ) |
11 |
2 3
|
reprfz1 |
⊢ ( 𝜑 → ( ℕ ( repr ‘ 𝑆 ) 𝑀 ) = ( ( 1 ... 𝑀 ) ( repr ‘ 𝑆 ) 𝑀 ) ) |
12 |
11
|
sumeq1d |
⊢ ( 𝜑 → Σ 𝑐 ∈ ( ℕ ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) = Σ 𝑐 ∈ ( ( 1 ... 𝑀 ) ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |
13 |
8 10 12
|
3eqtr4d |
⊢ ( 𝜑 → ( ♯ ‘ ( 𝐴 ( repr ‘ 𝑆 ) 𝑀 ) ) = Σ 𝑐 ∈ ( ℕ ( repr ‘ 𝑆 ) 𝑀 ) ∏ 𝑎 ∈ ( 0 ..^ 𝑆 ) ( ( ( 𝟭 ‘ ℕ ) ‘ 𝐴 ) ‘ ( 𝑐 ‘ 𝑎 ) ) ) |