Theorem tanhvaltan

Description: Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh . (Contributed by David A. Wheeler, 10-May-2015) (Revised by Jon Pennant, 22-Oct-2026)

	Ref	Expression
Assertion	tanhvaltan	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \tanh \left(A\right)=\frac{\tan \mathrm{i}A}{\mathrm{i}}$$

Proof

Step	Hyp	Ref	Expression
1		sinhval-named	$$⊢A\in ℂ\to \sinh \left(A\right)=\frac{\sin \mathrm{i}A}{\mathrm{i}}$$
2	1	adantr	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \sinh \left(A\right)=\frac{\sin \mathrm{i}A}{\mathrm{i}}$$
3		coshval-named	$$⊢A\in ℂ\to \cosh \left(A\right)=\cos \mathrm{i}A$$
4	3	adantr	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \cosh \left(A\right)=\cos \mathrm{i}A$$
5	2 4	oveq12d	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \frac{\sinh \left(A\right)}{\cosh \left(A\right)}=\frac{\frac{\sin \mathrm{i}A}{\mathrm{i}}}{\cos \mathrm{i}A}$$
6		tanhval	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \tanh \left(A\right)=\frac{\sinh \left(A\right)}{\cosh \left(A\right)}$$
7		ax-icn	$$⊢\mathrm{i}\in ℂ$$
8	7	a1i	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \mathrm{i}\in ℂ$$
9		simpl	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to A\in ℂ$$
10	8 9	mulcld	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \mathrm{i}A\in ℂ$$
11	10	sincld	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \sin \mathrm{i}A\in ℂ$$
12	10	coscld	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \cos \mathrm{i}A\in ℂ$$
13		simpr	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \cosh \left(A\right)\ne 0$$
14	4 13	eqnetrrd	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \cos \mathrm{i}A\ne 0$$
15		ine0	$$⊢\mathrm{i}\ne 0$$
16	15	a1i	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \mathrm{i}\ne 0$$
17	11 12 8 14 16	divdiv32d	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \frac{\frac{\sin \mathrm{i}A}{\cos \mathrm{i}A}}{\mathrm{i}}=\frac{\frac{\sin \mathrm{i}A}{\mathrm{i}}}{\cos \mathrm{i}A}$$
18	5 6 17	3eqtr4d	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \tanh \left(A\right)=\frac{\frac{\sin \mathrm{i}A}{\cos \mathrm{i}A}}{\mathrm{i}}$$
19		tanval	$$⊢\left(\mathrm{i}A\in ℂ\wedge \cos \mathrm{i}A\ne 0\right)\to \tan \mathrm{i}A=\frac{\sin \mathrm{i}A}{\cos \mathrm{i}A}$$
20	10 14 19	syl2anc	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \tan \mathrm{i}A=\frac{\sin \mathrm{i}A}{\cos \mathrm{i}A}$$
21	20	oveq1d	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \frac{\tan \mathrm{i}A}{\mathrm{i}}=\frac{\frac{\sin \mathrm{i}A}{\cos \mathrm{i}A}}{\mathrm{i}}$$
22	18 21	eqtr4d	$$⊢\left(A\in ℂ\wedge \cosh \left(A\right)\ne 0\right)\to \tanh \left(A\right)=\frac{\tan \mathrm{i}A}{\mathrm{i}}$$