Theorem rngonegcl

Description: A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010)

	Ref	Expression
Hypotheses	ringnegcl.1	$$⊢G=1^{st}\left(R\right)$$
	ringnegcl.2	$$⊢X=\mathrm{ran}G$$
	ringnegcl.3	$$⊢N=\mathrm{inv}\left(G\right)$$
Assertion	rngonegcl	$$⊢\left(R\in \mathrm{RingOps}\wedge A\in X\right)\to N\left(A\right)\in X$$

Proof

Step	Hyp	Ref	Expression
1		ringnegcl.1	$$⊢G=1^{st}\left(R\right)$$
2		ringnegcl.2	$$⊢X=\mathrm{ran}G$$
3		ringnegcl.3	$$⊢N=\mathrm{inv}\left(G\right)$$
4	1	rngogrpo	$$⊢R\in \mathrm{RingOps}\to G\in \mathrm{GrpOp}$$
5	2 3	grpoinvcl	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\right)\to N\left(A\right)\in X$$
6	4 5	sylan	$$⊢\left(R\in \mathrm{RingOps}\wedge A\in X\right)\to N\left(A\right)\in X$$