Step |
Hyp |
Ref |
Expression |
1 |
|
eldifsn |
⊢ ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ↔ ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1 ) ) |
2 |
|
rpcn |
⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ∈ ℂ ) |
3 |
2
|
adantr |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1 ) → 𝐵 ∈ ℂ ) |
4 |
|
rpne0 |
⊢ ( 𝐵 ∈ ℝ+ → 𝐵 ≠ 0 ) |
5 |
4
|
adantr |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1 ) → 𝐵 ≠ 0 ) |
6 |
|
simpr |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1 ) → 𝐵 ≠ 1 ) |
7 |
3 5 6
|
3jca |
⊢ ( ( 𝐵 ∈ ℝ+ ∧ 𝐵 ≠ 1 ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
8 |
1 7
|
sylbi |
⊢ ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
9 |
|
eldifpr |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
10 |
8 9
|
sylibr |
⊢ ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
11 |
10
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ∧ 𝐴 ∈ ℝ+ ) → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
12 |
|
simpr |
⊢ ( ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ∧ 𝐴 ∈ ℝ+ ) → 𝐴 ∈ ℝ+ ) |
13 |
|
eldifi |
⊢ ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) → 𝐵 ∈ ℝ+ ) |
14 |
13
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ∧ 𝐴 ∈ ℝ+ ) → 𝐵 ∈ ℝ+ ) |
15 |
|
relogbdiv |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+ ) ) → ( 𝐵 logb ( 𝐴 / 𝐵 ) ) = ( ( 𝐵 logb 𝐴 ) − ( 𝐵 logb 𝐵 ) ) ) |
16 |
11 12 14 15
|
syl12anc |
⊢ ( ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ∧ 𝐴 ∈ ℝ+ ) → ( 𝐵 logb ( 𝐴 / 𝐵 ) ) = ( ( 𝐵 logb 𝐴 ) − ( 𝐵 logb 𝐵 ) ) ) |
17 |
|
logbid1 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( 𝐵 logb 𝐵 ) = 1 ) |
18 |
8 17
|
syl |
⊢ ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) → ( 𝐵 logb 𝐵 ) = 1 ) |
19 |
18
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ∧ 𝐴 ∈ ℝ+ ) → ( 𝐵 logb 𝐵 ) = 1 ) |
20 |
19
|
oveq2d |
⊢ ( ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ∧ 𝐴 ∈ ℝ+ ) → ( ( 𝐵 logb 𝐴 ) − ( 𝐵 logb 𝐵 ) ) = ( ( 𝐵 logb 𝐴 ) − 1 ) ) |
21 |
16 20
|
eqtrd |
⊢ ( ( 𝐵 ∈ ( ℝ+ ∖ { 1 } ) ∧ 𝐴 ∈ ℝ+ ) → ( 𝐵 logb ( 𝐴 / 𝐵 ) ) = ( ( 𝐵 logb 𝐴 ) − 1 ) ) |