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Metamath Proof Explorer

Theorem hvsubeq0

Description: If the difference between two vectors is zero, they are equal. (Contributed by NM, 23-Oct-1999) (New usage is discouraged.)

Ref Expression
Assertion hvsubeq0 ⊢ A ∈ ℋ ∧ B ∈ ℋ → A - ℎ B = 0 ℎ ↔ A = B

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ A = if A ∈ ℋ A 0 ℎ → A - ℎ B = if A ∈ ℋ A 0 ℎ - ℎ B
2 1 eqeq1d ⊢ A = if A ∈ ℋ A 0 ℎ → A - ℎ B = 0 ℎ ↔ if A ∈ ℋ A 0 ℎ - ℎ B = 0 ℎ
3 eqeq1 ⊢ A = if A ∈ ℋ A 0 ℎ → A = B ↔ if A ∈ ℋ A 0 ℎ = B
4 2 3 bibi12d ⊢ A = if A ∈ ℋ A 0 ℎ → A - ℎ B = 0 ℎ ↔ A = B ↔ if A ∈ ℋ A 0 ℎ - ℎ B = 0 ℎ ↔ if A ∈ ℋ A 0 ℎ = B
5 oveq2 ⊢ B = if B ∈ ℋ B 0 ℎ → if A ∈ ℋ A 0 ℎ - ℎ B = if A ∈ ℋ A 0 ℎ - ℎ if B ∈ ℋ B 0 ℎ
6 5 eqeq1d ⊢ B = if B ∈ ℋ B 0 ℎ → if A ∈ ℋ A 0 ℎ - ℎ B = 0 ℎ ↔ if A ∈ ℋ A 0 ℎ - ℎ if B ∈ ℋ B 0 ℎ = 0 ℎ
7 eqeq2 ⊢ B = if B ∈ ℋ B 0 ℎ → if A ∈ ℋ A 0 ℎ = B ↔ if A ∈ ℋ A 0 ℎ = if B ∈ ℋ B 0 ℎ
8 6 7 bibi12d ⊢ B = if B ∈ ℋ B 0 ℎ → if A ∈ ℋ A 0 ℎ - ℎ B = 0 ℎ ↔ if A ∈ ℋ A 0 ℎ = B ↔ if A ∈ ℋ A 0 ℎ - ℎ if B ∈ ℋ B 0 ℎ = 0 ℎ ↔ if A ∈ ℋ A 0 ℎ = if B ∈ ℋ B 0 ℎ
9 ifhvhv0 ⊢ if A ∈ ℋ A 0 ℎ ∈ ℋ
10 ifhvhv0 ⊢ if B ∈ ℋ B 0 ℎ ∈ ℋ
11 9 10 hvsubeq0i ⊢ if A ∈ ℋ A 0 ℎ - ℎ if B ∈ ℋ B 0 ℎ = 0 ℎ ↔ if A ∈ ℋ A 0 ℎ = if B ∈ ℋ B 0 ℎ
12 4 8 11 dedth2h ⊢ A ∈ ℋ ∧ B ∈ ℋ → A - ℎ B = 0 ℎ ↔ A = B