Theorem rpcoshcl

Description: The hyperbolic cosine of a real number is a positive real. (Contributed by Mario Carneiro, 4-Apr-2015)

	Ref	Expression
Assertion	rpcoshcl	$$⊢A\in ℝ\to \cos \mathrm{i}A\in {ℝ}^{+}$$

Proof

Step	Hyp	Ref	Expression
1		recn	$$⊢A\in ℝ\to A\in ℂ$$
2		coshval	$$⊢A\in ℂ\to \cos \mathrm{i}A=\frac{{\mathrm{e}}^A+{\mathrm{e}}^{-A}}{2}$$
3	1 2	syl	$$⊢A\in ℝ\to \cos \mathrm{i}A=\frac{{\mathrm{e}}^A+{\mathrm{e}}^{-A}}{2}$$
4		rpefcl	$$⊢A\in ℝ\to {\mathrm{e}}^A\in {ℝ}^{+}$$
5		renegcl	$$⊢A\in ℝ\to -A\in ℝ$$
6	5	rpefcld	$$⊢A\in ℝ\to {\mathrm{e}}^{-A}\in {ℝ}^{+}$$
7	4 6	rpaddcld	$$⊢A\in ℝ\to {\mathrm{e}}^A+{\mathrm{e}}^{-A}\in {ℝ}^{+}$$
8	7	rphalfcld	$$⊢A\in ℝ\to \frac{{\mathrm{e}}^A+{\mathrm{e}}^{-A}}{2}\in {ℝ}^{+}$$
9	3 8	eqeltrd	$$⊢A\in ℝ\to \cos \mathrm{i}A\in {ℝ}^{+}$$