Step |
Hyp |
Ref |
Expression |
1 |
|
rpcn |
⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ ) |
2 |
1
|
adantr |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) → 𝐵 ∈ ℂ ) |
3 |
|
rpne0 |
⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ≠ 0 ) |
4 |
3
|
adantr |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) → 𝐵 ≠ 0 ) |
5 |
|
1red |
⊢ ( 𝐵 ∈ ℝ+ → 1 ∈ ℝ ) |
6 |
|
ltne |
⊢ ( ( 1 ∈ ℝ ∧ 1 < 𝐵 ) → 𝐵 ≠ 1 ) |
7 |
5 6
|
sylan |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) → 𝐵 ≠ 1 ) |
8 |
|
eldifpr |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
9 |
2 4 7 8
|
syl3anbrc |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
10 |
|
rpcndif0 |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ( ℂ ∖ { 0 } ) ) |
11 |
|
logbval |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐴 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 logb 𝐴 ) = ( ( log ‘ 𝐴 ) / ( log ‘ 𝐵 ) ) ) |
12 |
9 10 11
|
syl2anr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → ( 𝐵 logb 𝐴 ) = ( ( log ‘ 𝐴 ) / ( log ‘ 𝐵 ) ) ) |
13 |
12
|
breq2d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → ( 0 < ( 𝐵 logb 𝐴 ) ↔ 0 < ( ( log ‘ 𝐴 ) / ( log ‘ 𝐵 ) ) ) ) |
14 |
|
relogcl |
⊢ ( 𝐴 ∈ ℝ+ → ( log ‘ 𝐴 ) ∈ ℝ ) |
15 |
14
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → ( log ‘ 𝐴 ) ∈ ℝ ) |
16 |
|
relogcl |
⊢ ( 𝐵 ∈ ℝ+ → ( log ‘ 𝐵 ) ∈ ℝ ) |
17 |
16
|
adantr |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) → ( log ‘ 𝐵 ) ∈ ℝ ) |
18 |
17
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → ( log ‘ 𝐵 ) ∈ ℝ ) |
19 |
|
loggt0b |
⊢ ( 𝐵 ∈ ℝ+ → ( 0 < ( log ‘ 𝐵 ) ↔ 1 < 𝐵 ) ) |
20 |
19
|
biimpar |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) → 0 < ( log ‘ 𝐵 ) ) |
21 |
20
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → 0 < ( log ‘ 𝐵 ) ) |
22 |
|
gt0div |
⊢ ( ( ( log ‘ 𝐴 ) ∈ ℝ ∧ ( log ‘ 𝐵 ) ∈ ℝ ∧ 0 < ( log ‘ 𝐵 ) ) → ( 0 < ( log ‘ 𝐴 ) ↔ 0 < ( ( log ‘ 𝐴 ) / ( log ‘ 𝐵 ) ) ) ) |
23 |
15 18 21 22
|
syl3anc |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → ( 0 < ( log ‘ 𝐴 ) ↔ 0 < ( ( log ‘ 𝐴 ) / ( log ‘ 𝐵 ) ) ) ) |
24 |
|
loggt0b |
⊢ ( 𝐴 ∈ ℝ+ → ( 0 < ( log ‘ 𝐴 ) ↔ 1 < 𝐴 ) ) |
25 |
24
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → ( 0 < ( log ‘ 𝐴 ) ↔ 1 < 𝐴 ) ) |
26 |
13 23 25
|
3bitr2d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ ( 𝐵 ∈ ℝ+ ∧ 1 < 𝐵 ) ) → ( 0 < ( 𝐵 logb 𝐴 ) ↔ 1 < 𝐴 ) ) |