Step |
Hyp |
Ref |
Expression |
1 |
|
eldifpr |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
2 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
3 |
|
ax-1ne0 |
⊢ 1 ≠ 0 |
4 |
|
eldifsn |
⊢ ( 1 ∈ ( ℂ ∖ { 0 } ) ↔ ( 1 ∈ ℂ ∧ 1 ≠ 0 ) ) |
5 |
2 3 4
|
mpbir2an |
⊢ 1 ∈ ( ℂ ∖ { 0 } ) |
6 |
|
logbval |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 1 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 logb 1 ) = ( ( log ‘ 1 ) / ( log ‘ 𝐵 ) ) ) |
7 |
5 6
|
mpan2 |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( 𝐵 logb 1 ) = ( ( log ‘ 1 ) / ( log ‘ 𝐵 ) ) ) |
8 |
1 7
|
sylbir |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( 𝐵 logb 1 ) = ( ( log ‘ 1 ) / ( log ‘ 𝐵 ) ) ) |
9 |
|
log1 |
⊢ ( log ‘ 1 ) = 0 |
10 |
9
|
oveq1i |
⊢ ( ( log ‘ 1 ) / ( log ‘ 𝐵 ) ) = ( 0 / ( log ‘ 𝐵 ) ) |
11 |
|
simp1 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → 𝐵 ∈ ℂ ) |
12 |
|
simp2 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → 𝐵 ≠ 0 ) |
13 |
11 12
|
logcld |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( log ‘ 𝐵 ) ∈ ℂ ) |
14 |
|
logccne0 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( log ‘ 𝐵 ) ≠ 0 ) |
15 |
13 14
|
div0d |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( 0 / ( log ‘ 𝐵 ) ) = 0 ) |
16 |
10 15
|
syl5eq |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( ( log ‘ 1 ) / ( log ‘ 𝐵 ) ) = 0 ) |
17 |
8 16
|
eqtrd |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( 𝐵 logb 1 ) = 0 ) |