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Metamath Proof Explorer

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 ) ) )