Theorem recxpf1lem

Description: Complex exponentiation on positive real numbers is a one-to-one function. (Contributed by Thierry Arnoux, 1-Apr-2025)

	Ref	Expression
Hypotheses	recxpf1.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	recxpf1.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
	recxpf1.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
	recxpf1.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
	recxpf1.5	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
Assertion	recxpf1lem	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) = ( 𝐵 ↑𝑐 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		recxpf1.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		recxpf1.2	⊢ ( 𝜑 → 0 ≤ 𝐴 )
3		recxpf1.3	⊢ ( 𝜑 → 𝐵 ∈ ℝ )
4		recxpf1.4	⊢ ( 𝜑 → 0 ≤ 𝐵 )
5		recxpf1.5	⊢ ( 𝜑 → 𝐶 ∈ ℝ+ )
6	1 2 3 4 5	cxple2d	⊢ ( 𝜑 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) ) )
7	3 4 1 2 5	cxple2d	⊢ ( 𝜑 → ( 𝐵 ≤ 𝐴 ↔ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) )
8	6 7	anbi12d	⊢ ( 𝜑 → ( ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ↔ ( ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) ∧ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) ) )
9	1 3	letri3d	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐴 ) ) )
10	5	rpred	⊢ ( 𝜑 → 𝐶 ∈ ℝ )
11	1 2 10	recxpcld	⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐶 ) ∈ ℝ )
12	3 4 10	recxpcld	⊢ ( 𝜑 → ( 𝐵 ↑𝑐 𝐶 ) ∈ ℝ )
13	11 12	letri3d	⊢ ( 𝜑 → ( ( 𝐴 ↑𝑐 𝐶 ) = ( 𝐵 ↑𝑐 𝐶 ) ↔ ( ( 𝐴 ↑𝑐 𝐶 ) ≤ ( 𝐵 ↑𝑐 𝐶 ) ∧ ( 𝐵 ↑𝑐 𝐶 ) ≤ ( 𝐴 ↑𝑐 𝐶 ) ) ) )
14	8 9 13	3bitr4d	⊢ ( 𝜑 → ( 𝐴 = 𝐵 ↔ ( 𝐴 ↑𝑐 𝐶 ) = ( 𝐵 ↑𝑐 𝐶 ) ) )