Theorem hvaddid2t 5003

Description: Addition with the zero vector.

Assertion

Ref	Expression
hvaddid2t	⊢ (A ∈ ℋ → (0v +v A) = A)

Proof of Theorem hvaddid2t

Step	Hyp	Ref	Expression
1		ax-hvzercl 4987	. . 3 ⊢ 0v ∈ ℋ
2		ax-hvcom 4985	. . 3 ⊢ ((A ∈ ℋ ∧ 0v ∈ ℋ ) → (A +v 0v) = (0v +v A))
3	1, 2	mpan2 519	. 2 ⊢ (A ∈ ℋ → (A +v 0v) = (0v +v A))
4		ax-hvaddid 4988	. 2 ⊢ (A ∈ ℋ → (A +v 0v) = A)
5	3, 4	eqtr3d 1130	1 ⊢ (A ∈ ℋ → (0v +v A) = A)

Syntax hints:   → wi 2   = wceq 1091   ∈ wcel 1092  (class class class)co 3001   ℋ chil 4958   +_v cva 4959  0_vc0v 4961

This theorem is referenced by:  hvaddid2 5008  chocuni 5179  shunss 5338  spanunsn 5482  5oalem2 5545  3oalem2 5553

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677  ax-17 925  ax-ext 1074  ax-hvcom 4985  ax-hvzercl 4987  ax-hvaddid 4988

This theorem depends on definitions:  df-bi 128  df-an 198  df-cleq 1097

metamath.org