Step |
Hyp |
Ref |
Expression |
1 |
|
rxp11d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ+ ) |
2 |
|
rxp11d.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ+ ) |
3 |
|
rxp11d.3 |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
4 |
|
rxp11d.4 |
⊢ ( 𝜑 → 𝐶 ≠ 0 ) |
5 |
|
rxp11d.5 |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐶 ) = ( 𝐵 ↑𝑐 𝐶 ) ) |
6 |
1
|
relogcld |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℝ ) |
7 |
6
|
recnd |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) ∈ ℂ ) |
8 |
2
|
relogcld |
⊢ ( 𝜑 → ( log ‘ 𝐵 ) ∈ ℝ ) |
9 |
8
|
recnd |
⊢ ( 𝜑 → ( log ‘ 𝐵 ) ∈ ℂ ) |
10 |
3
|
recnd |
⊢ ( 𝜑 → 𝐶 ∈ ℂ ) |
11 |
5
|
fveq2d |
⊢ ( 𝜑 → ( log ‘ ( 𝐴 ↑𝑐 𝐶 ) ) = ( log ‘ ( 𝐵 ↑𝑐 𝐶 ) ) ) |
12 |
1 3
|
logcxpd |
⊢ ( 𝜑 → ( log ‘ ( 𝐴 ↑𝑐 𝐶 ) ) = ( 𝐶 · ( log ‘ 𝐴 ) ) ) |
13 |
2 3
|
logcxpd |
⊢ ( 𝜑 → ( log ‘ ( 𝐵 ↑𝑐 𝐶 ) ) = ( 𝐶 · ( log ‘ 𝐵 ) ) ) |
14 |
11 12 13
|
3eqtr3d |
⊢ ( 𝜑 → ( 𝐶 · ( log ‘ 𝐴 ) ) = ( 𝐶 · ( log ‘ 𝐵 ) ) ) |
15 |
7 9 10 4 14
|
mulcanad |
⊢ ( 𝜑 → ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ) |
16 |
1 2
|
rplog11d |
⊢ ( 𝜑 → ( ( log ‘ 𝐴 ) = ( log ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |
17 |
15 16
|
mpbid |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |