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Metamath Proof Explorer

Theorem hvsubid

Description: Subtraction of a vector from itself. (Contributed by NM, 30-May-1999) (New usage is discouraged.)

Ref Expression
Assertion hvsubid 
|- ( A e. ~H -> ( A -h A ) = 0h )

Proof

Step Hyp Ref Expression
1 ax-hvmulid 
 |-  ( A e. ~H -> ( 1 .h A ) = A )
2 1 oveq1d 
 |-  ( A e. ~H -> ( ( 1 .h A ) +h ( -u 1 .h A ) ) = ( A +h ( -u 1 .h A ) ) )
3 ax-1cn 
 |-  1 e. CC
4 neg1cn 
 |-  -u 1 e. CC
5 ax-hvdistr2 
 |-  ( ( 1 e. CC /\ -u 1 e. CC /\ A e. ~H ) -> ( ( 1 + -u 1 ) .h A ) = ( ( 1 .h A ) +h ( -u 1 .h A ) ) )
6 3 4 5 mp3an12 
 |-  ( A e. ~H -> ( ( 1 + -u 1 ) .h A ) = ( ( 1 .h A ) +h ( -u 1 .h A ) ) )
7 hvsubval 
 |-  ( ( A e. ~H /\ A e. ~H ) -> ( A -h A ) = ( A +h ( -u 1 .h A ) ) )
8 7 anidms 
 |-  ( A e. ~H -> ( A -h A ) = ( A +h ( -u 1 .h A ) ) )
9 2 6 8 3eqtr4rd 
 |-  ( A e. ~H -> ( A -h A ) = ( ( 1 + -u 1 ) .h A ) )
10 1pneg1e0 
 |-  ( 1 + -u 1 ) = 0
11 10 oveq1i 
 |-  ( ( 1 + -u 1 ) .h A ) = ( 0 .h A )
12 9 11 eqtrdi 
 |-  ( A e. ~H -> ( A -h A ) = ( 0 .h A ) )
13 ax-hvmul0 
 |-  ( A e. ~H -> ( 0 .h A ) = 0h )
14 12 13 eqtrd 
 |-  ( A e. ~H -> ( A -h A ) = 0h )