Theorem vcablo

Description: Vector addition is an Abelian group operation. (Contributed by NM, 3-Nov-2006) (New usage is discouraged.)

	Ref	Expression
Hypothesis	vcabl.1	⊢ 𝐺 = ( 1st ‘ 𝑊 )
Assertion	vcablo	⊢ ( 𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp )

Proof

Step	Hyp	Ref	Expression
1		vcabl.1	⊢ 𝐺 = ( 1st ‘ 𝑊 )
2		eqid	⊢ ( 2nd ‘ 𝑊 ) = ( 2nd ‘ 𝑊 )
3		eqid	⊢ ran 𝐺 = ran 𝐺
4	1 2 3	vciOLD	⊢ ( 𝑊 ∈ CVecOLD → ( 𝐺 ∈ AbelOp ∧ ( 2nd ‘ 𝑊 ) : ( ℂ × ran 𝐺 ) ⟶ ran 𝐺 ∧ ∀ 𝑥 ∈ ran 𝐺 ( ( 1 ( 2nd ‘ 𝑊 ) 𝑥 ) = 𝑥 ∧ ∀ 𝑦 ∈ ℂ ( ∀ 𝑧 ∈ ran 𝐺 ( 𝑦 ( 2nd ‘ 𝑊 ) ( 𝑥 𝐺 𝑧 ) ) = ( ( 𝑦 ( 2nd ‘ 𝑊 ) 𝑥 ) 𝐺 ( 𝑦 ( 2nd ‘ 𝑊 ) 𝑧 ) ) ∧ ∀ 𝑧 ∈ ℂ ( ( ( 𝑦 + 𝑧 ) ( 2nd ‘ 𝑊 ) 𝑥 ) = ( ( 𝑦 ( 2nd ‘ 𝑊 ) 𝑥 ) 𝐺 ( 𝑧 ( 2nd ‘ 𝑊 ) 𝑥 ) ) ∧ ( ( 𝑦 · 𝑧 ) ( 2nd ‘ 𝑊 ) 𝑥 ) = ( 𝑦 ( 2nd ‘ 𝑊 ) ( 𝑧 ( 2nd ‘ 𝑊 ) 𝑥 ) ) ) ) ) ) )
5	4	simp1d	⊢ ( 𝑊 ∈ CVecOLD → 𝐺 ∈ AbelOp )