Step |
Hyp |
Ref |
Expression |
1 |
|
df-blen |
⊢ #b = ( 𝑛 ∈ V ↦ if ( 𝑛 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑛 ) ) ) + 1 ) ) ) |
2 |
|
eqeq1 |
⊢ ( 𝑛 = 𝑁 → ( 𝑛 = 0 ↔ 𝑁 = 0 ) ) |
3 |
|
fveq2 |
⊢ ( 𝑛 = 𝑁 → ( abs ‘ 𝑛 ) = ( abs ‘ 𝑁 ) ) |
4 |
3
|
oveq2d |
⊢ ( 𝑛 = 𝑁 → ( 2 logb ( abs ‘ 𝑛 ) ) = ( 2 logb ( abs ‘ 𝑁 ) ) ) |
5 |
4
|
fveq2d |
⊢ ( 𝑛 = 𝑁 → ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑛 ) ) ) = ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) ) |
6 |
5
|
oveq1d |
⊢ ( 𝑛 = 𝑁 → ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑛 ) ) ) + 1 ) = ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ) |
7 |
2 6
|
ifbieq2d |
⊢ ( 𝑛 = 𝑁 → if ( 𝑛 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑛 ) ) ) + 1 ) ) = if ( 𝑁 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ) ) |
8 |
|
elex |
⊢ ( 𝑁 ∈ 𝑉 → 𝑁 ∈ V ) |
9 |
|
1ex |
⊢ 1 ∈ V |
10 |
|
ovex |
⊢ ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ∈ V |
11 |
9 10
|
ifex |
⊢ if ( 𝑁 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ) ∈ V |
12 |
11
|
a1i |
⊢ ( 𝑁 ∈ 𝑉 → if ( 𝑁 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ) ∈ V ) |
13 |
1 7 8 12
|
fvmptd3 |
⊢ ( 𝑁 ∈ 𝑉 → ( #b ‘ 𝑁 ) = if ( 𝑁 = 0 , 1 , ( ( ⌊ ‘ ( 2 logb ( abs ‘ 𝑁 ) ) ) + 1 ) ) ) |