Theorem rngorcan

Description: Right cancellation law for the addition operation of a ring. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ringgcl.1	$$⊢G=1^{st}\left(R\right)$$
	ringgcl.2	$$⊢X=\mathrm{ran}G$$
Assertion	rngorcan	$$⊢\left(R\in \mathrm{RingOps}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGC=BGC\leftrightarrow A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		ringgcl.1	$$⊢G=1^{st}\left(R\right)$$
2		ringgcl.2	$$⊢X=\mathrm{ran}G$$
3	1	rngogrpo	$$⊢R\in \mathrm{RingOps}\to G\in \mathrm{GrpOp}$$
4	2	grporcan	$$⊢\left(G\in \mathrm{GrpOp}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGC=BGC\leftrightarrow A=B\right)$$
5	3 4	sylan	$$⊢\left(R\in \mathrm{RingOps}\wedge \left(A\in X\wedge B\in X\wedge C\in X\right)\right)\to \left(AGC=BGC\leftrightarrow A=B\right)$$