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Theorem hvadd32i 22509
Description: Hilbert vector space commutative/associative law. (Contributed by NM, 18-Aug-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
hvadd32i  |-  ( ( A  +h  B )  +h  C )  =  ( ( A  +h  C )  +h  B
)

Proof of Theorem hvadd32i
StepHypRef Expression
1 hvass.1 . 2  |-  A  e. 
~H
2 hvass.2 . 2  |-  B  e. 
~H
3 hvass.3 . 2  |-  C  e. 
~H
4 hvadd32 22489 . 2  |-  ( ( A  e.  ~H  /\  B  e.  ~H  /\  C  e.  ~H )  ->  (
( A  +h  B
)  +h  C )  =  ( ( A  +h  C )  +h  B ) )
51, 2, 3, 4mp3an 1279 1  |-  ( ( A  +h  B )  +h  C )  =  ( ( A  +h  C )  +h  B
)
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721  (class class class)co 6040   ~Hchil 22375    +h cva 22376
This theorem is referenced by:  hvsubeq0i  22518  hvaddcani  22520  normpar2i  22611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-hvcom 22457  ax-hvass 22458
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-iota 5377  df-fv 5421  df-ov 6043
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