Theorem hvcomi 31005

Description: Commutation of vector addition. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hvaddcl.1	⊢ 𝐴 ∈ ℋ
hvaddcl.2	⊢ 𝐵 ∈ ℋ

Assertion

Ref	Expression
hvcomi	⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)

Proof of Theorem hvcomi

Step	Hyp	Ref	Expression
1		hvaddcl.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hvaddcl.2	. 2 ⊢ 𝐵 ∈ ℋ
3		ax-hvcom 30987	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))
4	1, 2, 3	mp2an 692	1 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)

Syntax hints:   = wceq 1540   ∈ wcel 2109  (class class class)co 7410   ℋchba 30905   +_ℎ cva 30906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvcom 30987

This theorem depends on definitions:  df-bi 207  df-an 396

This theorem is referenced by:  hvadd12i  31043  hvnegdii  31048  norm3difi  31133  normpar2i  31142  nonbooli  31637  lnophmlem2  32003