Theorem hvaddid2 27880

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

Step	Hyp	Ref	Expression
1		ax-hv0cl 27860	. . 3 |- 0h e. ~H
2		ax-hvcom 27858	. . 3 |- ( ( A e. ~H /\ 0h e. ~H ) -> ( A +h 0h ) = ( 0h +h A ) )
3	1, 2	mpan2 707	. 2 |- ( A e. ~H -> ( A +h 0h ) = ( 0h +h A ) )
4		ax-hvaddid 27861	. 2 |- ( A e. ~H -> ( A +h 0h ) = A )
5	3, 4	eqtr3d 2658	1 |- ( A e. ~H -> ( 0h +h A ) = A )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990  (class class class)co 6650   ~Hchil 27776   +h cva 27777   0hc0v 27781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602  ax-hvcom 27858  ax-hv0cl 27860  ax-hvaddid 27861

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615

This theorem is referenced by:  hv2neg  27885  hvaddid2i  27886  hvaddsub4  27935  hilablo  28017  hilid  28018  shunssi  28227  spanunsni  28438  5oalem2  28514  3oalem2  28522