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Theorem hvaddid2 25602
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 25582 . . 3  |-  0h  e.  ~H
2 ax-hvcom 25580 . . 3  |-  ( ( A  e.  ~H  /\  0h  e.  ~H )  -> 
( A  +h  0h )  =  ( 0h  +h  A ) )
31, 2mpan2 671 . 2  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  ( 0h  +h  A
) )
4 ax-hvaddid 25583 . 2  |-  ( A  e.  ~H  ->  ( A  +h  0h )  =  A )
53, 4eqtr3d 2503 1  |-  ( A  e.  ~H  ->  ( 0h  +h  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374    e. wcel 1762  (class class class)co 6275   ~Hchil 25498    +h cva 25499   0hc0v 25503
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-ext 2438  ax-hvcom 25580  ax-hv0cl 25582  ax-hvaddid 25583
This theorem depends on definitions:  df-bi 185  df-an 371  df-cleq 2452
This theorem is referenced by:  hv2neg  25607  hvaddid2i  25608  hvaddsub4  25657  hilablo  25739  hilid  25740  shunssi  25948  spanunsni  26159  5oalem2  26235  3oalem2  26243
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