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Metamath Proof Explorer

Theorem hvaddsubval

Description: Value of vector addition in terms of vector subtraction. (Contributed by NM, 10-Jun-2006) (New usage is discouraged.)

Ref Expression
Assertion hvaddsubval ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐴 −ℎ ( - 1 ·ℎ 𝐵 ) ) )

Proof

Step Hyp Ref Expression
1 neg1cn ⊢ - 1 ∈ ℂ
2 hvmulcl ⊢ ( ( - 1 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ 𝐵 ) ∈ ℋ )
3 1 2 mpan ⊢ ( 𝐵 ∈ ℋ → ( - 1 ·ℎ 𝐵 ) ∈ ℋ )
4 hvsubval ⊢ ( ( 𝐴 ∈ ℋ ∧ ( - 1 ·ℎ 𝐵 ) ∈ ℋ ) → ( 𝐴 −ℎ ( - 1 ·ℎ 𝐵 ) ) = ( 𝐴 +ℎ ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) ) )
5 3 4 sylan2 ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 −ℎ ( - 1 ·ℎ 𝐵 ) ) = ( 𝐴 +ℎ ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) ) )
6 hvm1neg ⊢ ( ( - 1 ∈ ℂ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) = ( - - 1 ·ℎ 𝐵 ) )
7 1 6 mpan ⊢ ( 𝐵 ∈ ℋ → ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) = ( - - 1 ·ℎ 𝐵 ) )
8 negneg1e1 ⊢ - - 1 = 1
9 8 oveq1i ⊢ ( - - 1 ·ℎ 𝐵 ) = ( 1 ·ℎ 𝐵 )
10 7 9 eqtrdi ⊢ ( 𝐵 ∈ ℋ → ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) = ( 1 ·ℎ 𝐵 ) )
11 ax-hvmulid ⊢ ( 𝐵 ∈ ℋ → ( 1 ·ℎ 𝐵 ) = 𝐵 )
12 10 11 eqtrd ⊢ ( 𝐵 ∈ ℋ → ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) = 𝐵 )
13 12 adantl ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) = 𝐵 )
14 13 oveq2d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ ( - 1 ·ℎ ( - 1 ·ℎ 𝐵 ) ) ) = ( 𝐴 +ℎ 𝐵 ) )
15 5 14 eqtr2d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 +ℎ 𝐵 ) = ( 𝐴 −ℎ ( - 1 ·ℎ 𝐵 ) ) )