Theorem hvass 5025

Description: Hilbert vector space associative law.

Hypotheses

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

Assertion

Ref	Expression
hvass	⊢ ((A +v B) +v C) = (A +v (B +v C))

Proof of Theorem hvass

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 4986	. 2 ⊢ ((A ∈ ℋ ∧ B ∈ ℋ ∧ C ∈ ℋ ) → ((A +v B) +v C) = (A +v (B +v C)))
5	1, 2, 3, 4	mp3an 642	1 ⊢ ((A +v B) +v C) = (A +v (B +v C))

Syntax hints:   = wceq 1091   ∈ wcel 1092  (class class class)co 3001   ℋ chil 4958   +_v cva 4959

This theorem is referenced by:  hvsubass 5027  hvadd12 5029  hvsubeq0 5035  norm3dif 5094

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-hvass 4986

This theorem depends on definitions:  df-bi 128  df-an 198  df-3an 583

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