Theorem resinhcl

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

	Ref	Expression
Assertion	resinhcl	$$⊢A\in ℝ\to \frac{\sin \mathrm{i}A}{\mathrm{i}}\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		recn	$$⊢A\in ℝ\to A\in ℂ$$
2		sinhval	$$⊢A\in ℂ\to \frac{\sin \mathrm{i}A}{\mathrm{i}}=\frac{{\mathrm{e}}^A-{\mathrm{e}}^{-A}}{2}$$
3	1 2	syl	$$⊢A\in ℝ\to \frac{\sin \mathrm{i}A}{\mathrm{i}}=\frac{{\mathrm{e}}^A-{\mathrm{e}}^{-A}}{2}$$
4		reefcl	$$⊢A\in ℝ\to {\mathrm{e}}^A\in ℝ$$
5		renegcl	$$⊢A\in ℝ\to -A\in ℝ$$
6	5	reefcld	$$⊢A\in ℝ\to {\mathrm{e}}^{-A}\in ℝ$$
7	4 6	resubcld	$$⊢A\in ℝ\to {\mathrm{e}}^A-{\mathrm{e}}^{-A}\in ℝ$$
8	7	rehalfcld	$$⊢A\in ℝ\to \frac{{\mathrm{e}}^A-{\mathrm{e}}^{-A}}{2}\in ℝ$$
9	3 8	eqeltrd	$$⊢A\in ℝ\to \frac{\sin \mathrm{i}A}{\mathrm{i}}\in ℝ$$