Metamath Proof Explorer

Theorem hlipgt0

Description: The inner product of a Hilbert space vector by itself is positive. (Contributed by NM, 8-Sep-2007) (New usage is discouraged.)

Ref Expression
Hypotheses hlipgt0.1 ⊢ X = BaseSet ⁡ U
hlipgt0.5 ⊢ Z = 0 vec ⁡ U
hlipgt0.7 ⊢ P = ⋅ 𝑖OLD ⁡ U
Assertion hlipgt0 ⊢ U ∈ CHil OLD ∧ A ∈ X ∧ A ≠ Z → 0 < A P A

Proof

Step Hyp Ref Expression
1 hlipgt0.1 ⊢ X = BaseSet ⁡ U
2 hlipgt0.5 ⊢ Z = 0 vec ⁡ U
3 hlipgt0.7 ⊢ P = ⋅ 𝑖OLD ⁡ U
4 hlnv ⊢ U ∈ CHil OLD → U ∈ NrmCVec
5 eqid ⊢ norm CV ⁡ U = norm CV ⁡ U
6 1 5 nvcl ⊢ U ∈ NrmCVec ∧ A ∈ X → norm CV ⁡ U ⁡ A ∈ ℝ
7 6 3adant3 ⊢ U ∈ NrmCVec ∧ A ∈ X ∧ A ≠ Z → norm CV ⁡ U ⁡ A ∈ ℝ
8 1 2 5 nvz ⊢ U ∈ NrmCVec ∧ A ∈ X → norm CV ⁡ U ⁡ A = 0 ↔ A = Z
9 8 biimpd ⊢ U ∈ NrmCVec ∧ A ∈ X → norm CV ⁡ U ⁡ A = 0 → A = Z
10 9 necon3d ⊢ U ∈ NrmCVec ∧ A ∈ X → A ≠ Z → norm CV ⁡ U ⁡ A ≠ 0
11 10 3impia ⊢ U ∈ NrmCVec ∧ A ∈ X ∧ A ≠ Z → norm CV ⁡ U ⁡ A ≠ 0
12 7 11 sqgt0d ⊢ U ∈ NrmCVec ∧ A ∈ X ∧ A ≠ Z → 0 < norm CV ⁡ U ⁡ A 2
13 1 5 3 ipidsq ⊢ U ∈ NrmCVec ∧ A ∈ X → A P A = norm CV ⁡ U ⁡ A 2
14 13 3adant3 ⊢ U ∈ NrmCVec ∧ A ∈ X ∧ A ≠ Z → A P A = norm CV ⁡ U ⁡ A 2
15 12 14 breqtrrd ⊢ U ∈ NrmCVec ∧ A ∈ X ∧ A ≠ Z → 0 < A P A
16 4 15 syl3an1 ⊢ U ∈ CHil OLD ∧ A ∈ X ∧ A ≠ Z → 0 < A P A