Theorem ghmgrp2

Description: A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014)

	Ref	Expression
Assertion	ghmgrp2	$$⊢F\in \left(S\mathrm{GrpHom}T\right)\to T\in \mathrm{Grp}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{\mathrm{Base}}_S={\mathrm{Base}}_S$$
2		eqid	$$⊢{\mathrm{Base}}_T={\mathrm{Base}}_T$$
3		eqid	$$⊢{+}_S={+}_S$$
4		eqid	$$⊢{+}_T={+}_T$$
5	1 2 3 4	isghm	$$⊢F\in \left(S\mathrm{GrpHom}T\right)\leftrightarrow \left(\left(S\in \mathrm{Grp}\wedge T\in \mathrm{Grp}\right)\wedge \left(F:{\mathrm{Base}}_S⟶{\mathrm{Base}}_T\wedge \forall y\in {\mathrm{Base}}_S\forall x\in {\mathrm{Base}}_{S}F\left(y{+}_{S}x\right)=F\left(y\right){+}_{T}F\left(x\right)\right)\right)$$
6	5	simplbi	$$⊢F\in \left(S\mathrm{GrpHom}T\right)\to \left(S\in \mathrm{Grp}\wedge T\in \mathrm{Grp}\right)$$
7	6	simprd	$$⊢F\in \left(S\mathrm{GrpHom}T\right)\to T\in \mathrm{Grp}$$