Theorem rngo0lid

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

	Ref	Expression
Hypotheses	ring0cl.1	⊢ 𝐺 = ( 1st ‘ 𝑅 )
	ring0cl.2	⊢ 𝑋 = ran 𝐺
	ring0cl.3	⊢ 𝑍 = ( GId ‘ 𝐺 )
Assertion	rngo0lid	⊢ ( ( 𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋 ) → ( 𝑍 𝐺 𝐴 ) = 𝐴 )

Proof

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