Theorem vc0rid

Description: The zero vector is a right identity element. (Contributed by NM, 4-Nov-2006) (New usage is discouraged.)

	Ref	Expression
Hypotheses	vczcl.1	⊢ 𝐺 = ( 1st ‘ 𝑊 )
	vczcl.2	⊢ 𝑋 = ran 𝐺
	vczcl.3	⊢ 𝑍 = ( GId ‘ 𝐺 )
Assertion	vc0rid	⊢ ( ( 𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		vczcl.1	⊢ 𝐺 = ( 1st ‘ 𝑊 )
2		vczcl.2	⊢ 𝑋 = ran 𝐺
3		vczcl.3	⊢ 𝑍 = ( GId ‘ 𝐺 )
4	1	vcgrp	⊢ ( 𝑊 ∈ CVecOLD → 𝐺 ∈ GrpOp )
5	2 3	grporid	⊢ ( ( 𝐺 ∈ GrpOp ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )
6	4 5	sylan	⊢ ( ( 𝑊 ∈ CVecOLD ∧ 𝐴 ∈ 𝑋 ) → ( 𝐴 𝐺 𝑍 ) = 𝐴 )