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Metamath Proof Explorer

Theorem hvsubcan2

Description: Cancellation law for vector addition. (Contributed by NM, 18-May-2005) (New usage is discouraged.)

Ref Expression
Assertion hvsubcan2 ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐶 ) = ( 𝐵 −ℎ 𝐶 ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 hvsubcl ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐶 −ℎ 𝐴 ) ∈ ℋ )
2 1 3adant3 ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐶 −ℎ 𝐴 ) ∈ ℋ )
3 hvsubcl ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐶 −ℎ 𝐵 ) ∈ ℋ )
4 3 3adant2 ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐶 −ℎ 𝐵 ) ∈ ℋ )
5 neg1cn ⊢ - 1 ∈ ℂ
6 neg1ne0 ⊢ - 1 ≠ 0
7 5 6 pm3.2i ⊢ ( - 1 ∈ ℂ ∧ - 1 ≠ 0 )
8 hvmulcan ⊢ ( ( ( - 1 ∈ ℂ ∧ - 1 ≠ 0 ) ∧ ( 𝐶 −ℎ 𝐴 ) ∈ ℋ ∧ ( 𝐶 −ℎ 𝐵 ) ∈ ℋ ) → ( ( - 1 ·ℎ ( 𝐶 −ℎ 𝐴 ) ) = ( - 1 ·ℎ ( 𝐶 −ℎ 𝐵 ) ) ↔ ( 𝐶 −ℎ 𝐴 ) = ( 𝐶 −ℎ 𝐵 ) ) )
9 7 8 mp3an1 ⊢ ( ( ( 𝐶 −ℎ 𝐴 ) ∈ ℋ ∧ ( 𝐶 −ℎ 𝐵 ) ∈ ℋ ) → ( ( - 1 ·ℎ ( 𝐶 −ℎ 𝐴 ) ) = ( - 1 ·ℎ ( 𝐶 −ℎ 𝐵 ) ) ↔ ( 𝐶 −ℎ 𝐴 ) = ( 𝐶 −ℎ 𝐵 ) ) )
10 2 4 9 syl2anc ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( - 1 ·ℎ ( 𝐶 −ℎ 𝐴 ) ) = ( - 1 ·ℎ ( 𝐶 −ℎ 𝐵 ) ) ↔ ( 𝐶 −ℎ 𝐴 ) = ( 𝐶 −ℎ 𝐵 ) ) )
11 hvnegdi ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ) → ( - 1 ·ℎ ( 𝐶 −ℎ 𝐴 ) ) = ( 𝐴 −ℎ 𝐶 ) )
12 11 3adant3 ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( 𝐶 −ℎ 𝐴 ) ) = ( 𝐴 −ℎ 𝐶 ) )
13 hvnegdi ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( 𝐶 −ℎ 𝐵 ) ) = ( 𝐵 −ℎ 𝐶 ) )
14 13 3adant2 ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( - 1 ·ℎ ( 𝐶 −ℎ 𝐵 ) ) = ( 𝐵 −ℎ 𝐶 ) )
15 12 14 eqeq12d ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( - 1 ·ℎ ( 𝐶 −ℎ 𝐴 ) ) = ( - 1 ·ℎ ( 𝐶 −ℎ 𝐵 ) ) ↔ ( 𝐴 −ℎ 𝐶 ) = ( 𝐵 −ℎ 𝐶 ) ) )
16 hvsubcan ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐶 −ℎ 𝐴 ) = ( 𝐶 −ℎ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
17 10 15 16 3bitr3d ⊢ ( ( 𝐶 ∈ ℋ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐶 ) = ( 𝐵 −ℎ 𝐶 ) ↔ 𝐴 = 𝐵 ) )
18 17 3coml ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐶 ) = ( 𝐵 −ℎ 𝐶 ) ↔ 𝐴 = 𝐵 ) )