Theorem hvass 8920

Description: Hilbert vector space associative law.

Hypotheses

Ref	Expression
hvass.1	|- A e. H~
hvass.2	|- B e. H~
hvass.3	|- C e. H~

Assertion

Ref	Expression
hvass	|- ((A +h B) +h C) = (A +h (B +h C))

Proof of Theorem hvass

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 8872	. 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 916	1 |- ((A +h B) +h C) = (A +h (B +h C))

Syntax hints:   = wceq 956   e. wcel 958  (class class class)co 3963  H~chil 8788   +h cva 8789

This theorem is referenced by:  hvsubass 8922  hvadd12 8924  hvsubeq0 8930  norm3dif 9014

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-hvass 8872

This theorem depends on definitions:  df-bi 147  df-an 225  df-3an 777

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