Theorem rhmrcl1

Description: Reverse closure of a ring homomorphism. (Contributed by Stefan O'Rear, 7-Mar-2015)

	Ref	Expression
Assertion	rhmrcl1	$$⊢F\in \left(R\mathrm{RingHom}S\right)\to R\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		dfrhm2	$$⊢\mathrm{RingHom}=\left(r\in \mathrm{Ring},s\in \mathrm{Ring}⟼\left(r\mathrm{GrpHom}s\right)\cap \left({\mathrm{mulGrp}}_r\mathrm{MndHom}{\mathrm{mulGrp}}_s\right)\right)$$
2	1	elmpocl1	$$⊢F\in \left(R\mathrm{RingHom}S\right)\to R\in \mathrm{Ring}$$