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Theorem hvassi 28840
 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 28789 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
51, 2, 3, 4mp3an 1458 1 ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2112  (class class class)co 7139   ℋchba 28706   +ℎ cva 28707 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvass 28789 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086 This theorem is referenced by:  hvadd12i  28844  hvsubeq0i  28850  norm3difi  28934
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