Step |
Hyp |
Ref |
Expression |
1 |
|
rxp112d.c |
⊢ ( 𝜑 → 𝐶 ∈ ℝ+ ) |
2 |
|
rxp112d.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
3 |
|
rxp112d.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
rxp112d.1 |
⊢ ( 𝜑 → 𝐶 ≠ 1 ) |
5 |
|
rxp112d.2 |
⊢ ( 𝜑 → ( 𝐶 ↑𝑐 𝐴 ) = ( 𝐶 ↑𝑐 𝐵 ) ) |
6 |
2
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
7 |
3
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
8 |
1
|
relogcld |
⊢ ( 𝜑 → ( log ‘ 𝐶 ) ∈ ℝ ) |
9 |
8
|
recnd |
⊢ ( 𝜑 → ( log ‘ 𝐶 ) ∈ ℂ ) |
10 |
1 4
|
logne0d |
⊢ ( 𝜑 → ( log ‘ 𝐶 ) ≠ 0 ) |
11 |
5
|
fveq2d |
⊢ ( 𝜑 → ( log ‘ ( 𝐶 ↑𝑐 𝐴 ) ) = ( log ‘ ( 𝐶 ↑𝑐 𝐵 ) ) ) |
12 |
1 2
|
logcxpd |
⊢ ( 𝜑 → ( log ‘ ( 𝐶 ↑𝑐 𝐴 ) ) = ( 𝐴 · ( log ‘ 𝐶 ) ) ) |
13 |
1 3
|
logcxpd |
⊢ ( 𝜑 → ( log ‘ ( 𝐶 ↑𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐶 ) ) ) |
14 |
11 12 13
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝐴 · ( log ‘ 𝐶 ) ) = ( 𝐵 · ( log ‘ 𝐶 ) ) ) |
15 |
6 7 9 10 14
|
mulcan2ad |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |