Theorem srngcnv

Description: The involution function in a star ring is its own inverse function. (Contributed by Mario Carneiro, 6-Oct-2015)

	Ref	Expression
Hypothesis	srngcnv.i	$$⊢\underset{\dot{}}{\ast }={\ast }_{\mathrm{𝑟𝑓}}\left(R\right)$$
Assertion	srngcnv	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }={\underset{\dot{}}{\ast }}^{-1}$$

Proof

Step	Hyp	Ref	Expression
1		srngcnv.i	$$⊢\underset{\dot{}}{\ast }={\ast }_{\mathrm{𝑟𝑓}}\left(R\right)$$
2		eqid	$$⊢{opp}_r\left(R\right)={opp}_r\left(R\right)$$
3	2 1	issrng	$$⊢R\in \mathrm{*-Ring}\leftrightarrow \left(\underset{\dot{}}{\ast }\in \left(R\mathrm{RingHom}{opp}_r\left(R\right)\right)\wedge \underset{\dot{}}{\ast }={\underset{\dot{}}{\ast }}^{-1}\right)$$
4	3	simprbi	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }={\underset{\dot{}}{\ast }}^{-1}$$