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 ) ) |