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Theorem hvadd12i 21691
Description: Hilbert vector space commutative/associative law. (Contributed by NM, 11-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
hvadd12i  |-  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) )

Proof of Theorem hvadd12i
StepHypRef Expression
1 hvass.1 . . . 4  |-  A  e. 
~H
2 hvass.2 . . . 4  |-  B  e. 
~H
31, 2hvcomi 21654 . . 3  |-  ( A  +h  B )  =  ( B  +h  A
)
43oveq1i 5910 . 2  |-  ( ( A  +h  B )  +h  C )  =  ( ( B  +h  A )  +h  C
)
5 hvass.3 . . 3  |-  C  e. 
~H
61, 2, 5hvassi 21687 . 2  |-  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) )
72, 1, 5hvassi 21687 . 2  |-  ( ( B  +h  A )  +h  C )  =  ( B  +h  ( A  +h  C ) )
84, 6, 73eqtr3i 2344 1  |-  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1633    e. wcel 1701  (class class class)co 5900   ~Hchil 21554    +h cva 21555
This theorem is referenced by:  hvsubaddi  21700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-hvcom 21636  ax-hvass 21637
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-rex 2583  df-rab 2586  df-v 2824  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903
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