Theorem srngring

Description: A star ring is a ring. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Assertion	srngring	$$⊢R\in \mathrm{*-Ring}\to R\in \mathrm{Ring}$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢{opp}_r\left(R\right)={opp}_r\left(R\right)$$
2		eqid	$$⊢{\ast }_{\mathrm{𝑟𝑓}}\left(R\right)={\ast }_{\mathrm{𝑟𝑓}}\left(R\right)$$
3	1 2	srngrhm	$$⊢R\in \mathrm{*-Ring}\to {\ast }_{\mathrm{𝑟𝑓}}\left(R\right)\in \left(R\mathrm{RingHom}{opp}_r\left(R\right)\right)$$
4		rhmrcl1	$$⊢{\ast }_{\mathrm{𝑟𝑓}}\left(R\right)\in \left(R\mathrm{RingHom}{opp}_r\left(R\right)\right)\to R\in \mathrm{Ring}$$
5	3 4	syl	$$⊢R\in \mathrm{*-Ring}\to R\in \mathrm{Ring}$$