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Theorem hvassi 27294
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 27243 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
51, 2, 3, 4mp3an 1416 1 ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wcel 1977  (class class class)co 6549  chil 27160   + cva 27161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvass 27243
This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033
This theorem is referenced by:  hvadd12i  27298  hvsubeq0i  27304  norm3difi  27388
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