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Theorem hvadd12i 21636
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 21599 . . 3  |-  ( A  +h  B )  =  ( B  +h  A
)
43oveq1i 5868 . 2  |-  ( ( A  +h  B )  +h  C )  =  ( ( B  +h  A )  +h  C
)
5 hvass.3 . . 3  |-  C  e. 
~H
61, 2, 5hvassi 21632 . 2  |-  ( ( A  +h  B )  +h  C )  =  ( A  +h  ( B  +h  C ) )
72, 1, 5hvassi 21632 . 2  |-  ( ( B  +h  A )  +h  C )  =  ( B  +h  ( A  +h  C ) )
84, 6, 73eqtr3i 2311 1  |-  ( A  +h  ( B  +h  C ) )  =  ( B  +h  ( A  +h  C ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. wcel 1684  (class class class)co 5858   ~Hchil 21499    +h cva 21500
This theorem is referenced by:  hvsubaddi  21645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-hvcom 21581  ax-hvass 21582
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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