Theorem hvcomi 31444

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 31426	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐴 +ℎ 𝐵) = (𝐵 +ℎ 𝐴)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  (class class class)co 7419   ℋchba 31344   +_ℎ cva 31345

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvcom 31426

This proof depends on definitions:  df-bi 210  df-an 402

This theorem is used by:  hvadd12i  31482  hvnegdii  31487  norm3difi  31572  normpar2i  31581  nonbooli  32076  lnophmlem2  32442