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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 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
rxp11d.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ+ )
rxp11d.3 ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
rxp11d.4 ⊢ ( 𝜑 → 𝐶 ≠ 0 )
rxp11d.5 ⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐶 ) = ( 𝐵 ↑𝑐 𝐶 ) )
Assertion rxp11d ( 𝜑 → 𝐴 = 𝐵 )

Proof

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 )