Metamath Proof Explorer


Theorem rxp112d

Description: Real exponentiation is one-to-one with respect to the second argument. (TODO: Note that the base C must be positive since -u C ^ A is C ^ A x.e ^ i _pi A , so in the negative case A = B + 2 k ). (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses rxp112d.c ( 𝜑𝐶 ∈ ℝ+ )
rxp112d.a ( 𝜑𝐴 ∈ ℝ )
rxp112d.b ( 𝜑𝐵 ∈ ℝ )
rxp112d.1 ( 𝜑𝐶 ≠ 1 )
rxp112d.2 ( 𝜑 → ( 𝐶𝑐 𝐴 ) = ( 𝐶𝑐 𝐵 ) )
Assertion rxp112d ( 𝜑𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 rxp112d.c ( 𝜑𝐶 ∈ ℝ+ )
2 rxp112d.a ( 𝜑𝐴 ∈ ℝ )
3 rxp112d.b ( 𝜑𝐵 ∈ ℝ )
4 rxp112d.1 ( 𝜑𝐶 ≠ 1 )
5 rxp112d.2 ( 𝜑 → ( 𝐶𝑐 𝐴 ) = ( 𝐶𝑐 𝐵 ) )
6 2 recnd ( 𝜑𝐴 ∈ ℂ )
7 3 recnd ( 𝜑𝐵 ∈ 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 ‘ ( 𝐶𝑐 𝐴 ) ) = ( log ‘ ( 𝐶𝑐 𝐵 ) ) )
12 1 2 logcxpd ( 𝜑 → ( log ‘ ( 𝐶𝑐 𝐴 ) ) = ( 𝐴 · ( log ‘ 𝐶 ) ) )
13 1 3 logcxpd ( 𝜑 → ( log ‘ ( 𝐶𝑐 𝐵 ) ) = ( 𝐵 · ( log ‘ 𝐶 ) ) )
14 11 12 13 3eqtr3d ( 𝜑 → ( 𝐴 · ( log ‘ 𝐶 ) ) = ( 𝐵 · ( log ‘ 𝐶 ) ) )
15 6 7 9 10 14 mulcan2ad ( 𝜑𝐴 = 𝐵 )