Theorem hvcomi 30199

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 30181	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))
4	1, 2, 3	mp2an 690	1 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)

Syntax hints:   = wceq 1541   ∈ wcel 2106  (class class class)co 7394   ℋchba 30099   +_ℎ cva 30100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvcom 30181

This theorem depends on definitions:  df-bi 206  df-an 397

This theorem is referenced by:  hvadd12i  30237  hvnegdii  30242  norm3difi  30327  normpar2i  30336  nonbooli  30831  lnophmlem2  31197