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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 = ( 0vec ‘ 𝑈 )
hlipgt0.7 ⊢ 𝑃 = ( ·𝑖OLD ‘ 𝑈 )
Assertion hlipgt0 ( ( 𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 𝑍 ) → 0 < ( 𝐴 𝑃 𝐴 ) )

Proof

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