Theorem hvaddlid 31682

Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)

Assertion

Ref	Expression
hvaddlid	⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

Proof of Theorem hvaddlid

Step	Hyp	Ref	Expression
1		ax-hv0cl 31662	. . 3 ⊢ 0ℎ ∈ ℋ
2		ax-hvcom 31660	. . 3 ⊢ ((𝐴 ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
3	1, 2	mpan2 704	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = (0ℎ +ℎ 𝐴))
4		ax-hvaddid 31663	. 2 ⊢ (𝐴 ∈ ℋ → (𝐴 +ℎ 0ℎ) = 𝐴)
5	3, 4	eqtr3d 2800	1 ⊢ (𝐴 ∈ ℋ → (0ℎ +ℎ 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  (class class class)co 7451   ℋchba 31578   +_ℎ cva 31579  0_ℎc0v 31583

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735  ax-hvcom 31660  ax-hv0cl 31662  ax-hvaddid 31663

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755

This theorem is used by:  hv2neg  31687  hvaddlidi  31688  hvaddsub4  31737  hilablo  31819  hilid  31820  shunssi  32027  spanunsni  32238  5oalem2  32314  3oalem2  32322