Theorem rngolcan

Description: Left 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 = ( 1st ‘ 𝑅 )
	ringgcl.2	⊢ 𝑋 = ran 𝐺
Assertion	rngolcan	⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 𝐺 𝐴 ) = ( 𝐶 𝐺 𝐵 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ringgcl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
2		ringgcl.2	⊢ 𝑋 = ran 𝐺
3	1	rngogrpo	⊢ ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
4	2	grpolcan	⊢ ( ( 𝐺 ∈ GrpOp ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 𝐺 𝐴 ) = ( 𝐶 𝐺 𝐵 ) ↔ 𝐴 = 𝐵 ) )
5	3 4	sylan	⊢ ( ( 𝑅 ∈ RingOps ∧ ( 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋 ) ) → ( ( 𝐶 𝐺 𝐴 ) = ( 𝐶 𝐺 𝐵 ) ↔ 𝐴 = 𝐵 ) )