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Metamath Proof Explorer

Theorem hv2times

Description: Two times a vector. (Contributed by NM, 22-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion hv2times 
|- ( A e. ~H -> ( 2 .h A ) = ( A +h A ) )

Proof

Step Hyp Ref Expression
1 df-2 
 |-  2 = ( 1 + 1 )
2 1 oveq1i 
 |-  ( 2 .h A ) = ( ( 1 + 1 ) .h A )
3 ax-1cn 
 |-  1 e. CC
4 ax-hvdistr2 
 |-  ( ( 1 e. CC /\ 1 e. CC /\ A e. ~H ) -> ( ( 1 + 1 ) .h A ) = ( ( 1 .h A ) +h ( 1 .h A ) ) )
5 3 3 4 mp3an12 
 |-  ( A e. ~H -> ( ( 1 + 1 ) .h A ) = ( ( 1 .h A ) +h ( 1 .h A ) ) )
6 2 5 eqtrid 
 |-  ( A e. ~H -> ( 2 .h A ) = ( ( 1 .h A ) +h ( 1 .h A ) ) )
7 ax-hvdistr1 
 |-  ( ( 1 e. CC /\ A e. ~H /\ A e. ~H ) -> ( 1 .h ( A +h A ) ) = ( ( 1 .h A ) +h ( 1 .h A ) ) )
8 3 7 mp3an1 
 |-  ( ( A e. ~H /\ A e. ~H ) -> ( 1 .h ( A +h A ) ) = ( ( 1 .h A ) +h ( 1 .h A ) ) )
9 8 anidms 
 |-  ( A e. ~H -> ( 1 .h ( A +h A ) ) = ( ( 1 .h A ) +h ( 1 .h A ) ) )
10 hvaddcl 
 |-  ( ( A e. ~H /\ A e. ~H ) -> ( A +h A ) e. ~H )
11 10 anidms 
 |-  ( A e. ~H -> ( A +h A ) e. ~H )
12 ax-hvmulid 
 |-  ( ( A +h A ) e. ~H -> ( 1 .h ( A +h A ) ) = ( A +h A ) )
13 11 12 syl 
 |-  ( A e. ~H -> ( 1 .h ( A +h A ) ) = ( A +h A ) )
14 6 9 13 3eqtr2d 
 |-  ( A e. ~H -> ( 2 .h A ) = ( A +h A ) )