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	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
	rxp112d.a	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	rxp112d.b	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	rxp112d.1	⊢ ( 𝜑 → 𝐶 ≠ 1 )
	rxp112d.2	⊢ ( 𝜑 → ( 𝐶 ↑𝑐 𝐴 ) = ( 𝐶 ↑𝑐 𝐵 ) )
Assertion	rxp112d	⊢ ( 𝜑 → 𝐴 = 𝐵 )

Proof

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 )