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 = ( 0vec𝑈 )
hlipgt0.7 𝑃 = ( ·𝑖OLD𝑈 )
Assertion hlipgt0 ( ( 𝑈 ∈ CHilOLD𝐴𝑋𝐴𝑍 ) → 0 < ( 𝐴 𝑃 𝐴 ) )

Proof

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