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Theorem hvassi 27880
 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 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))

Proof of Theorem hvassi
StepHypRef Expression
1 hvass.1 . 2 𝐴 ∈ ℋ
2 hvass.2 . 2 𝐵 ∈ ℋ
3 hvass.3 . 2 𝐶 ∈ ℋ
4 ax-hvass 27829 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
51, 2, 3, 4mp3an 1422 1 ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1481   ∈ wcel 1988  (class class class)co 6635   ℋchil 27746   +ℎ cva 27747 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvass 27829 This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1038 This theorem is referenced by:  hvadd12i  27884  hvsubeq0i  27890  norm3difi  27974
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