Theorem hvassi 31039

Description: Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hvass.1	⊢ 𝐴 ∈ ℋ
hvass.2	⊢ 𝐵 ∈ ℋ
hvass.3	⊢ 𝐶 ∈ ℋ

Assertion

Ref	Expression
hvassi	⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))

Proof of Theorem hvassi

Step	Hyp	Ref	Expression
1		hvass.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hvass.2	. 2 ⊢ 𝐵 ∈ ℋ
3		hvass.3	. 2 ⊢ 𝐶 ∈ ℋ
4		ax-hvass 30988	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶)))
5	1, 2, 3, 4	mp3an 1463	1 ⊢ ((𝐴 +ℎ 𝐵) +ℎ 𝐶) = (𝐴 +ℎ (𝐵 +ℎ 𝐶))

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-hvass 30988

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088

This theorem is referenced by:  hvadd12i  31043  hvsubeq0i  31049  norm3difi  31133