Metamath Proof Explorer

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 = 1 st ⁡ 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 = 1 st ⁡ 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