Theorem rngoaddneg1

Description: Adding the negative in a ring gives zero. (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	$$⊢N=\mathrm{inv}\left(G\right)$$
	ringaddneg.4	$$⊢Z=\mathrm{GId}\left(G\right)$$
Assertion	rngoaddneg1	$$⊢\left(R\in \mathrm{RingOps}\wedge A\in X\right)\to AGN\left(A\right)=Z$$

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		ringaddneg.4	$$⊢Z=\mathrm{GId}\left(G\right)$$
5	1	rngogrpo	$$⊢R\in \mathrm{RingOps}\to G\in \mathrm{GrpOp}$$
6	2 4 3	grporinv	$$⊢\left(G\in \mathrm{GrpOp}\wedge A\in X\right)\to AGN\left(A\right)=Z$$
7	5 6	sylan	$$⊢\left(R\in \mathrm{RingOps}\wedge A\in X\right)\to AGN\left(A\right)=Z$$