Step |
Hyp |
Ref |
Expression |
1 |
|
logbval |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 logb 𝑋 ) = ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) ) |
2 |
1
|
oveq2d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) = ( 𝐵 ↑𝑐 ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) ) ) |
3 |
|
eldifi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → 𝐵 ∈ ℂ ) |
4 |
3
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → 𝐵 ∈ ℂ ) |
5 |
|
eldif |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ { 0 , 1 } ) ) |
6 |
|
c0ex |
⊢ 0 ∈ V |
7 |
6
|
prid1 |
⊢ 0 ∈ { 0 , 1 } |
8 |
|
eleq1 |
⊢ ( 𝐵 = 0 → ( 𝐵 ∈ { 0 , 1 } ↔ 0 ∈ { 0 , 1 } ) ) |
9 |
7 8
|
mpbiri |
⊢ ( 𝐵 = 0 → 𝐵 ∈ { 0 , 1 } ) |
10 |
9
|
necon3bi |
⊢ ( ¬ 𝐵 ∈ { 0 , 1 } → 𝐵 ≠ 0 ) |
11 |
10
|
adantl |
⊢ ( ( 𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ { 0 , 1 } ) → 𝐵 ≠ 0 ) |
12 |
5 11
|
sylbi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → 𝐵 ≠ 0 ) |
13 |
12
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → 𝐵 ≠ 0 ) |
14 |
|
eldif |
⊢ ( 𝑋 ∈ ( ℂ ∖ { 0 } ) ↔ ( 𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ { 0 } ) ) |
15 |
6
|
snid |
⊢ 0 ∈ { 0 } |
16 |
|
eleq1 |
⊢ ( 𝑋 = 0 → ( 𝑋 ∈ { 0 } ↔ 0 ∈ { 0 } ) ) |
17 |
15 16
|
mpbiri |
⊢ ( 𝑋 = 0 → 𝑋 ∈ { 0 } ) |
18 |
17
|
necon3bi |
⊢ ( ¬ 𝑋 ∈ { 0 } → 𝑋 ≠ 0 ) |
19 |
18
|
anim2i |
⊢ ( ( 𝑋 ∈ ℂ ∧ ¬ 𝑋 ∈ { 0 } ) → ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) ) |
20 |
14 19
|
sylbi |
⊢ ( 𝑋 ∈ ( ℂ ∖ { 0 } ) → ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) ) |
21 |
|
logcl |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) → ( log ‘ 𝑋 ) ∈ ℂ ) |
22 |
20 21
|
syl |
⊢ ( 𝑋 ∈ ( ℂ ∖ { 0 } ) → ( log ‘ 𝑋 ) ∈ ℂ ) |
23 |
22
|
adantl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( log ‘ 𝑋 ) ∈ ℂ ) |
24 |
10
|
anim2i |
⊢ ( ( 𝐵 ∈ ℂ ∧ ¬ 𝐵 ∈ { 0 , 1 } ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) |
25 |
5 24
|
sylbi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) ) |
26 |
|
logcl |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ) → ( log ‘ 𝐵 ) ∈ ℂ ) |
27 |
25 26
|
syl |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( log ‘ 𝐵 ) ∈ ℂ ) |
28 |
27
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( log ‘ 𝐵 ) ∈ ℂ ) |
29 |
|
eldifpr |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ↔ ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
30 |
29
|
biimpi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
31 |
30
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) ) |
32 |
|
logccne0 |
⊢ ( ( 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0 ∧ 𝐵 ≠ 1 ) → ( log ‘ 𝐵 ) ≠ 0 ) |
33 |
31 32
|
syl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( log ‘ 𝐵 ) ≠ 0 ) |
34 |
23 28 33
|
divcld |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) ∈ ℂ ) |
35 |
4 13 34
|
cxpefd |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 ↑𝑐 ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) ) = ( exp ‘ ( ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) · ( log ‘ 𝐵 ) ) ) ) |
36 |
|
eldifsn |
⊢ ( 𝑋 ∈ ( ℂ ∖ { 0 } ) ↔ ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) ) |
37 |
36 21
|
sylbi |
⊢ ( 𝑋 ∈ ( ℂ ∖ { 0 } ) → ( log ‘ 𝑋 ) ∈ ℂ ) |
38 |
37
|
adantl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( log ‘ 𝑋 ) ∈ ℂ ) |
39 |
29 32
|
sylbi |
⊢ ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) → ( log ‘ 𝐵 ) ≠ 0 ) |
40 |
39
|
adantr |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( log ‘ 𝐵 ) ≠ 0 ) |
41 |
38 28 40
|
divcan1d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) · ( log ‘ 𝐵 ) ) = ( log ‘ 𝑋 ) ) |
42 |
41
|
fveq2d |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( exp ‘ ( ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) · ( log ‘ 𝐵 ) ) ) = ( exp ‘ ( log ‘ 𝑋 ) ) ) |
43 |
|
eflog |
⊢ ( ( 𝑋 ∈ ℂ ∧ 𝑋 ≠ 0 ) → ( exp ‘ ( log ‘ 𝑋 ) ) = 𝑋 ) |
44 |
36 43
|
sylbi |
⊢ ( 𝑋 ∈ ( ℂ ∖ { 0 } ) → ( exp ‘ ( log ‘ 𝑋 ) ) = 𝑋 ) |
45 |
44
|
adantl |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( exp ‘ ( log ‘ 𝑋 ) ) = 𝑋 ) |
46 |
42 45
|
eqtrd |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( exp ‘ ( ( ( log ‘ 𝑋 ) / ( log ‘ 𝐵 ) ) · ( log ‘ 𝐵 ) ) ) = 𝑋 ) |
47 |
2 35 46
|
3eqtrd |
⊢ ( ( 𝐵 ∈ ( ℂ ∖ { 0 , 1 } ) ∧ 𝑋 ∈ ( ℂ ∖ { 0 } ) ) → ( 𝐵 ↑𝑐 ( 𝐵 logb 𝑋 ) ) = 𝑋 ) |