Theorem vczcl

Description: The zero vector is a vector. (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	vczcl	⊢ ( 𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋 )

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	grpoidcl	⊢ ( 𝐺 ∈ GrpOp → 𝑍 ∈ 𝑋 )
6	4 5	syl	⊢ ( 𝑊 ∈ CVecOLD → 𝑍 ∈ 𝑋 )