Theorem srngrhm

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

	Ref	Expression
Hypotheses	issrng.o	$$⊢O={opp}_r\left(R\right)$$
	issrng.i	$$⊢\underset{\dot{}}{\ast }={\ast }_{\mathrm{𝑟𝑓}}\left(R\right)$$
Assertion	srngrhm	$$⊢R\in \mathrm{*-Ring}\to \underset{\dot{}}{\ast }\in \left(R\mathrm{RingHom}O\right)$$

Proof

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