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 ∈ ℋ → ( 𝐴 𝐴 ) = 0 )

Proof

Step Hyp Ref Expression
1 ax-hvmulid ( 𝐴 ∈ ℋ → ( 1 · 𝐴 ) = 𝐴 )
2 1 oveq1d ( 𝐴 ∈ ℋ → ( ( 1 · 𝐴 ) + ( - 1 · 𝐴 ) ) = ( 𝐴 + ( - 1 · 𝐴 ) ) )
3 ax-1cn 1 ∈ ℂ
4 neg1cn - 1 ∈ ℂ
5 ax-hvdistr2 ( ( 1 ∈ ℂ ∧ - 1 ∈ ℂ ∧ 𝐴 ∈ ℋ ) → ( ( 1 + - 1 ) · 𝐴 ) = ( ( 1 · 𝐴 ) + ( - 1 · 𝐴 ) ) )
6 3 4 5 mp3an12 ( 𝐴 ∈ ℋ → ( ( 1 + - 1 ) · 𝐴 ) = ( ( 1 · 𝐴 ) + ( - 1 · 𝐴 ) ) )
7 hvsubval ( ( 𝐴 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 𝐴 ) = ( 𝐴 + ( - 1 · 𝐴 ) ) )
8 7 anidms ( 𝐴 ∈ ℋ → ( 𝐴 𝐴 ) = ( 𝐴 + ( - 1 · 𝐴 ) ) )
9 2 6 8 3eqtr4rd ( 𝐴 ∈ ℋ → ( 𝐴 𝐴 ) = ( ( 1 + - 1 ) · 𝐴 ) )
10 1pneg1e0 ( 1 + - 1 ) = 0
11 10 oveq1i ( ( 1 + - 1 ) · 𝐴 ) = ( 0 · 𝐴 )
12 9 11 eqtrdi ( 𝐴 ∈ ℋ → ( 𝐴 𝐴 ) = ( 0 · 𝐴 ) )
13 ax-hvmul0 ( 𝐴 ∈ ℋ → ( 0 · 𝐴 ) = 0 )
14 12 13 eqtrd ( 𝐴 ∈ ℋ → ( 𝐴 𝐴 ) = 0 )