Theorem hvassi 27910

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

Step	Hyp	Ref	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 27859	. 2 |- ( ( A e. ~H /\ B e. ~H /\ C e. ~H ) -> ( ( A +h B ) +h C ) = ( A +h ( B +h C ) ) )
5	1, 2, 3, 4	mp3an 1424	1 |- ( ( A +h B ) +h C ) = ( A +h ( B +h C ) )

Syntax hints:   = wceq 1483   e. wcel 1990  (class class class)co 6650   ~Hchil 27776   +h cva 27777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-hvass 27859

This theorem depends on definitions:  df-bi 197  df-an 386  df-3an 1039

This theorem is referenced by:  hvadd12i  27914  hvsubeq0i  27920  norm3difi  28004