Step |
Hyp |
Ref |
Expression |
1 |
|
logcxp |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( log ‘ ( 𝐶 ↑𝑐 𝐸 ) ) = ( 𝐸 · ( log ‘ 𝐶 ) ) ) |
2 |
1
|
3adant1 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( log ‘ ( 𝐶 ↑𝑐 𝐸 ) ) = ( 𝐸 · ( log ‘ 𝐶 ) ) ) |
3 |
2
|
oveq1d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( ( log ‘ ( 𝐶 ↑𝑐 𝐸 ) ) / ( log ‘ 𝐵 ) ) = ( ( 𝐸 · ( log ‘ 𝐶 ) ) / ( log ‘ 𝐵 ) ) ) |
4 |
|
recn |
⊢ ( 𝐸 ∈ ℝ → 𝐸 ∈ ℂ ) |
5 |
4
|
3ad2ant3 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → 𝐸 ∈ ℂ ) |
6 |
|
rpcn |
⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ℂ ) |
7 |
|
rpne0 |
⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ≠ 0 ) |
8 |
6 7
|
logcld |
⊢ ( 𝐶 ∈ ℝ+ → ( log ‘ 𝐶 ) ∈ ℂ ) |
9 |
8
|
3ad2ant2 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( log ‘ 𝐶 ) ∈ ℂ ) |
10 |
|
eldifi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → 𝐵 ∈ ℂ ) |
11 |
|
eldifpr |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
12 |
11
|
simp2bi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → 𝐵 ≠ 0 ) |
13 |
10 12
|
logcld |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( log ‘ 𝐵 ) ∈ ℂ ) |
14 |
|
logccne0 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( log ‘ 𝐵 ) ≠ 0 ) |
15 |
11 14
|
sylbi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( log ‘ 𝐵 ) ≠ 0 ) |
16 |
13 15
|
jca |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( ( log ‘ 𝐵 ) ∈ ℂ ∧ ( log ‘ 𝐵 ) ≠ 0 ) ) |
17 |
16
|
3ad2ant1 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( ( log ‘ 𝐵 ) ∈ ℂ ∧ ( log ‘ 𝐵 ) ≠ 0 ) ) |
18 |
|
divass |
⊢ ( ( 𝐸 ∈ ℂ ∧ ( log ‘ 𝐶 ) ∈ ℂ ∧ ( ( log ‘ 𝐵 ) ∈ ℂ ∧ ( log ‘ 𝐵 ) ≠ 0 ) ) → ( ( 𝐸 · ( log ‘ 𝐶 ) ) / ( log ‘ 𝐵 ) ) = ( 𝐸 · ( ( log ‘ 𝐶 ) / ( log ‘ 𝐵 ) ) ) ) |
19 |
5 9 17 18
|
syl3anc |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( ( 𝐸 · ( log ‘ 𝐶 ) ) / ( log ‘ 𝐵 ) ) = ( 𝐸 · ( ( log ‘ 𝐶 ) / ( log ‘ 𝐵 ) ) ) ) |
20 |
3 19
|
eqtrd |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( ( log ‘ ( 𝐶 ↑𝑐 𝐸 ) ) / ( log ‘ 𝐵 ) ) = ( 𝐸 · ( ( log ‘ 𝐶 ) / ( log ‘ 𝐵 ) ) ) ) |
21 |
|
simp1 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ) |
22 |
6
|
adantr |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → 𝐶 ∈ ℂ ) |
23 |
4
|
adantl |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → 𝐸 ∈ ℂ ) |
24 |
22 23
|
cxpcld |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐶 ↑𝑐 𝐸 ) ∈ ℂ ) |
25 |
7
|
adantr |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → 𝐶 ≠ 0 ) |
26 |
22 25 23
|
cxpne0d |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐶 ↑𝑐 𝐸 ) ≠ 0 ) |
27 |
|
eldifsn |
⊢ ( ( 𝐶 ↑𝑐 𝐸 ) ∈ ( ℂ ∖ { 0 } ) ↔ ( ( 𝐶 ↑𝑐 𝐸 ) ∈ ℂ ∧ ( 𝐶 ↑𝑐 𝐸 ) ≠ 0 ) ) |
28 |
24 26 27
|
sylanbrc |
⊢ ( ( 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐶 ↑𝑐 𝐸 ) ∈ ( ℂ ∖ { 0 } ) ) |
29 |
28
|
3adant1 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐶 ↑𝑐 𝐸 ) ∈ ( ℂ ∖ { 0 } ) ) |
30 |
|
logbval |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ ( 𝐶 ↑𝑐 𝐸 ) ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 logb ( 𝐶 ↑𝑐 𝐸 ) ) = ( ( log ‘ ( 𝐶 ↑𝑐 𝐸 ) ) / ( log ‘ 𝐵 ) ) ) |
31 |
21 29 30
|
syl2anc |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐵 logb ( 𝐶 ↑𝑐 𝐸 ) ) = ( ( log ‘ ( 𝐶 ↑𝑐 𝐸 ) ) / ( log ‘ 𝐵 ) ) ) |
32 |
|
rpcndif0 |
⊢ ( 𝐶 ∈ ℝ+ → 𝐶 ∈ ( ℂ ∖ { 0 } ) ) |
33 |
32
|
anim2i |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ) → ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ( ℂ ∖ { 0 } ) ) ) |
34 |
33
|
3adant3 |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ( ℂ ∖ { 0 } ) ) ) |
35 |
|
logbval |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 logb 𝐶 ) = ( ( log ‘ 𝐶 ) / ( log ‘ 𝐵 ) ) ) |
36 |
34 35
|
syl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐵 logb 𝐶 ) = ( ( log ‘ 𝐶 ) / ( log ‘ 𝐵 ) ) ) |
37 |
36
|
oveq2d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐸 · ( 𝐵 logb 𝐶 ) ) = ( 𝐸 · ( ( log ‘ 𝐶 ) / ( log ‘ 𝐵 ) ) ) ) |
38 |
20 31 37
|
3eqtr4d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝐶 ∈ ℝ+ ∧ 𝐸 ∈ ℝ ) → ( 𝐵 logb ( 𝐶 ↑𝑐 𝐸 ) ) = ( 𝐸 · ( 𝐵 logb 𝐶 ) ) ) |