Metamath Proof Explorer

Theorem rngoaass

Description: Obsolete theorem, use ringgrp and grpass instead. The addition operation of a ring is associative. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.) (Proof modification is discouraged.)

Ref Expression
Hypotheses ringgcl.1 ⊢ G = 1 st ⁡ R
ringgcl.2 ⊢ X = ran ⁡ G
Assertion rngoaass ⊢ R ∈ RingOps ∧ A ∈ X ∧ B ∈ X ∧ C ∈ X → A G B G C = A G B G C

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 grpoass ⊢ G ∈ GrpOp ∧ A ∈ X ∧ B ∈ X ∧ C ∈ X → A G B G C = A G B G C
5 3 4 sylan ⊢ R ∈ RingOps ∧ A ∈ X ∧ B ∈ X ∧ C ∈ X → A G B G C = A G B G C