Metamath Proof Explorer


Theorem rngo0rid

Description: The additive identity of a ring is a right identity element. (Contributed by Steve Rodriguez, 9-Sep-2007) (New usage is discouraged.)

Ref Expression
Hypotheses ring0cl.1 𝐺 = ( 1st𝑅 )
ring0cl.2 𝑋 = ran 𝐺
ring0cl.3 𝑍 = ( GId ‘ 𝐺 )
Assertion rngo0rid ( ( 𝑅 ∈ RingOps ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )

Proof

Step Hyp Ref Expression
1 ring0cl.1 𝐺 = ( 1st𝑅 )
2 ring0cl.2 𝑋 = ran 𝐺
3 ring0cl.3 𝑍 = ( GId ‘ 𝐺 )
4 1 rngogrpo ( 𝑅 ∈ RingOps → 𝐺 ∈ GrpOp )
5 2 3 grporid ( ( 𝐺 ∈ GrpOp ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )
6 4 5 sylan ( ( 𝑅 ∈ RingOps ∧ 𝐴𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )