Step |
Hyp |
Ref |
Expression |
1 |
|
pjhcl |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ ) |
2 |
|
normcl |
⊢ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ℝ ) |
3 |
1 2
|
syl |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ∈ ℝ ) |
4 |
|
normcl |
⊢ ( 𝐴 ∈ ℋ → ( normℎ ‘ 𝐴 ) ∈ ℝ ) |
5 |
4
|
adantl |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ 𝐴 ) ∈ ℝ ) |
6 |
3 5
|
eqleltd |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ↔ ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ∧ ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ) ) |
7 |
|
pjnorm |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ) |
8 |
7
|
biantrurd |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ↔ ( ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) ≤ ( normℎ ‘ 𝐴 ) ∧ ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ) ) |
9 |
|
pjnel |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ) ) |
10 |
9
|
con1bid |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ¬ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) < ( normℎ ‘ 𝐴 ) ↔ 𝐴 ∈ 𝐻 ) ) |
11 |
6 8 10
|
3bitr2rd |
⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝐴 ∈ 𝐻 ↔ ( normℎ ‘ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( normℎ ‘ 𝐴 ) ) ) |