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Theorem hvassi 26171
Description: Hilbert vector space associative law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvass.1  |-  A  e. 
~H
hvass.2  |-  B  e. 
~H
hvass.3  |-  C  e. 
~H
Assertion
Ref Expression
hvassi  |-  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) )

Proof of Theorem hvassi
StepHypRef Expression
1 hvass.1 . 2  |-  A  e. 
~H
2 hvass.2 . 2  |-  B  e. 
~H
3 hvass.3 . 2  |-  C  e. 
~H
4 ax-hvass 26120 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( A  +h  ( B  +h  C
) ) )
51, 2, 3, 4mp3an 1322 1  |-  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398    e. wcel 1823  (class class class)co 6270   ~Hchil 26037    +h cva 26038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvass 26120
This theorem depends on definitions:  df-bi 185  df-an 369  df-3an 973
This theorem is referenced by:  hvadd12i  26175  hvsubeq0i  26181  norm3difi  26265
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