Metamath Proof Explorer


Theorem rxp11d

Description: Real exponentiation is one-to-one with respect to the first argument. (Contributed by SN, 25-Apr-2025)

Ref Expression
Hypotheses rxp11d.1 ( 𝜑𝐴 ∈ ℝ+ )
rxp11d.2 ( 𝜑𝐵 ∈ ℝ+ )
rxp11d.3 ( 𝜑𝐶 ∈ ℝ )
rxp11d.4 ( 𝜑𝐶 ≠ 0 )
rxp11d.5 ( 𝜑 → ( 𝐴𝑐 𝐶 ) = ( 𝐵𝑐 𝐶 ) )
Assertion rxp11d ( 𝜑𝐴 = 𝐵 )

Proof

Step Hyp Ref Expression
1 rxp11d.1 ( 𝜑𝐴 ∈ ℝ+ )
2 rxp11d.2 ( 𝜑𝐵 ∈ ℝ+ )
3 rxp11d.3 ( 𝜑𝐶 ∈ ℝ )
4 rxp11d.4 ( 𝜑𝐶 ≠ 0 )
5 rxp11d.5 ( 𝜑 → ( 𝐴𝑐 𝐶 ) = ( 𝐵𝑐 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 ( 𝜑𝐶 ∈ ℂ )
11 5 fveq2d ( 𝜑 → ( log ‘ ( 𝐴𝑐 𝐶 ) ) = ( log ‘ ( 𝐵𝑐 𝐶 ) ) )
12 1 3 logcxpd ( 𝜑 → ( log ‘ ( 𝐴𝑐 𝐶 ) ) = ( 𝐶 · ( log ‘ 𝐴 ) ) )
13 2 3 logcxpd ( 𝜑 → ( log ‘ ( 𝐵𝑐 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 ( 𝜑𝐴 = 𝐵 )