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Theorem hvaddid2 24360
Description: Addition with the zero vector. (Contributed by NM, 18-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvaddid2  |-  ( A  e.  ~H  ->  ( 0h  +h  A )  =  A )

Proof of Theorem hvaddid2
StepHypRef Expression
1 ax-hv0cl 24340 . . 3  |-  0h  e.  ~H
2 ax-hvcom 24338 . . 3  |-  ( ( A  e.  ~H  /\  0h  e.  ~H )  -> 
( A  +h  0h )  =  ( 0h  +h  A ) )
31, 2mpan2 666 . 2  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  ( 0h  +h  A
) )
4 ax-hvaddid 24341 . 2  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
53, 4eqtr3d 2475 1  |-  ( A  e.  ~H  ->  ( 0h  +h  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761  (class class class)co 6090   ~Hchil 24256    +h cva 24257   0hc0v 24261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-12 1797  ax-ext 2422  ax-hvcom 24338  ax-hv0cl 24340  ax-hvaddid 24341
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-cleq 2434
This theorem is referenced by:  hv2neg  24365  hvaddid2i  24366  hvaddsub4  24415  hilablo  24497  hilid  24498  shunssi  24706  spanunsni  24917  5oalem2  24993  3oalem2  25001
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