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	$$⊢\phi \to A\in ℝ$$
	recxpf1.2	$$⊢\phi \to 0\le A$$
	rexcpf1.3	$$⊢\phi \to B\in ℝ$$
	rexcpf1.4	$$⊢\phi \to 0\le B$$
	rexcpf1.5	$$⊢\phi \to C\in {ℝ}^{+}$$
Assertion	recxpf1lem	$$⊢\phi \to \left(A=B\leftrightarrow A^C=B^C\right)$$

Proof

Step	Hyp	Ref	Expression
1		recxpf1.1	$$⊢\phi \to A\in ℝ$$
2		recxpf1.2	$$⊢\phi \to 0\le A$$
3		rexcpf1.3	$$⊢\phi \to B\in ℝ$$
4		rexcpf1.4	$$⊢\phi \to 0\le B$$
5		rexcpf1.5	$$⊢\phi \to C\in {ℝ}^{+}$$
6	1 2 3 4 5	cxple2d	$$⊢\phi \to \left(A\le B\leftrightarrow A^C\le B^C\right)$$
7	3 4 1 2 5	cxple2d	$$⊢\phi \to \left(B\le A\leftrightarrow B^C\le A^C\right)$$
8	6 7	anbi12d	$$⊢\phi \to \left(\left(A\le B\wedge B\le A\right)\leftrightarrow \left(A^C\le B^C\wedge B^C\le A^C\right)\right)$$
9	1 3	letri3d	$$⊢\phi \to \left(A=B\leftrightarrow \left(A\le B\wedge B\le A\right)\right)$$
10	5	rpred	$$⊢\phi \to C\in ℝ$$
11	1 2 10	recxpcld	$$⊢\phi \to A^C\in ℝ$$
12	3 4 10	recxpcld	$$⊢\phi \to B^C\in ℝ$$
13	11 12	letri3d	$$⊢\phi \to \left(A^C=B^C\leftrightarrow \left(A^C\le B^C\wedge B^C\le A^C\right)\right)$$
14	8 9 13	3bitr4d	$$⊢\phi \to \left(A=B\leftrightarrow A^C=B^C\right)$$