Step |
Hyp |
Ref |
Expression |
1 |
|
neg1rr |
⊢ - 1 ∈ ℝ |
2 |
|
relogbmulexp |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ∧ - 1 ∈ ℝ ) ) → ( 𝐵 logb ( 𝐴 · ( 𝐶 ↑𝑐 - 1 ) ) ) = ( ( 𝐵 logb 𝐴 ) + ( - 1 · ( 𝐵 logb 𝐶 ) ) ) ) |
3 |
1 2
|
mp3anr3 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( 𝐵 logb ( 𝐴 · ( 𝐶 ↑𝑐 - 1 ) ) ) = ( ( 𝐵 logb 𝐴 ) + ( - 1 · ( 𝐵 logb 𝐶 ) ) ) ) |
4 |
|
rpcn |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ℂ ) |
5 |
4
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → 𝐴 ∈ ℂ ) |
6 |
|
rpcn |
⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ℂ ) |
7 |
6
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → 𝐶 ∈ ℂ ) |
8 |
|
rpne0 |
⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ≠ 0 ) |
9 |
8
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → 𝐶 ≠ 0 ) |
10 |
5 7 9
|
divrecd |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 / 𝐶 ) = ( 𝐴 · ( 1 / 𝐶 ) ) ) |
11 |
|
1cnd |
⊢ ( 𝐶 ∈ ℝ+ → 1 ∈ ℂ ) |
12 |
6 8 11
|
cxpnegd |
⊢ ( 𝐶 ∈ ℝ+ → ( 𝐶 ↑𝑐 - 1 ) = ( 1 / ( 𝐶 ↑𝑐 1 ) ) ) |
13 |
6
|
cxp1d |
⊢ ( 𝐶 ∈ ℝ+ → ( 𝐶 ↑𝑐 1 ) = 𝐶 ) |
14 |
13
|
oveq2d |
⊢ ( 𝐶 ∈ ℝ+ → ( 1 / ( 𝐶 ↑𝑐 1 ) ) = ( 1 / 𝐶 ) ) |
15 |
12 14
|
eqtrd |
⊢ ( 𝐶 ∈ ℝ+ → ( 𝐶 ↑𝑐 - 1 ) = ( 1 / 𝐶 ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → ( 𝐶 ↑𝑐 - 1 ) = ( 1 / 𝐶 ) ) |
17 |
16
|
oveq2d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 · ( 𝐶 ↑𝑐 - 1 ) ) = ( 𝐴 · ( 1 / 𝐶 ) ) ) |
18 |
10 17
|
eqtr4d |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → ( 𝐴 / 𝐶 ) = ( 𝐴 · ( 𝐶 ↑𝑐 - 1 ) ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( 𝐴 / 𝐶 ) = ( 𝐴 · ( 𝐶 ↑𝑐 - 1 ) ) ) |
20 |
19
|
oveq2d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( 𝐵 logb ( 𝐴 / 𝐶 ) ) = ( 𝐵 logb ( 𝐴 · ( 𝐶 ↑𝑐 - 1 ) ) ) ) |
21 |
|
rpcndif0 |
⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ( ℂ ∖ { 0 } ) ) |
22 |
21
|
adantl |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → 𝐶 ∈ ( ℂ ∖ { 0 } ) ) |
23 |
|
logbcl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 logb 𝐶 ) ∈ ℂ ) |
24 |
22 23
|
sylan2 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( 𝐵 logb 𝐶 ) ∈ ℂ ) |
25 |
|
mulm1 |
⊢ ( ( 𝐵 logb 𝐶 ) ∈ ℂ → ( - 1 · ( 𝐵 logb 𝐶 ) ) = - ( 𝐵 logb 𝐶 ) ) |
26 |
25
|
oveq2d |
⊢ ( ( 𝐵 logb 𝐶 ) ∈ ℂ → ( ( 𝐵 logb 𝐴 ) + ( - 1 · ( 𝐵 logb 𝐶 ) ) ) = ( ( 𝐵 logb 𝐴 ) + - ( 𝐵 logb 𝐶 ) ) ) |
27 |
24 26
|
syl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( ( 𝐵 logb 𝐴 ) + ( - 1 · ( 𝐵 logb 𝐶 ) ) ) = ( ( 𝐵 logb 𝐴 ) + - ( 𝐵 logb 𝐶 ) ) ) |
28 |
|
rpcndif0 |
⊢ ( 𝐴 ∈ ℝ+ → 𝐴 ∈ ( ℂ ∖ { 0 } ) ) |
29 |
28
|
adantr |
⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) → 𝐴 ∈ ( ℂ ∖ { 0 } ) ) |
30 |
|
logbcl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐴 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 logb 𝐴 ) ∈ ℂ ) |
31 |
29 30
|
sylan2 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( 𝐵 logb 𝐴 ) ∈ ℂ ) |
32 |
31 24
|
negsubd |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( ( 𝐵 logb 𝐴 ) + - ( 𝐵 logb 𝐶 ) ) = ( ( 𝐵 logb 𝐴 ) − ( 𝐵 logb 𝐶 ) ) ) |
33 |
27 32
|
eqtr2d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( ( 𝐵 logb 𝐴 ) − ( 𝐵 logb 𝐶 ) ) = ( ( 𝐵 logb 𝐴 ) + ( - 1 · ( 𝐵 logb 𝐶 ) ) ) ) |
34 |
3 20 33
|
3eqtr4d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐴 ∈ ℝ+ ∧ 𝐶 ∈ ℝ+ ) ) → ( 𝐵 logb ( 𝐴 / 𝐶 ) ) = ( ( 𝐵 logb 𝐴 ) − ( 𝐵 logb 𝐶 ) ) ) |