Step |
Hyp |
Ref |
Expression |
1 |
|
pjhcl |
|- ( ( H e. CH /\ A e. ~H ) -> ( ( projh ` H ) ` A ) e. ~H ) |
2 |
|
normcl |
|- ( ( ( projh ` H ) ` A ) e. ~H -> ( normh ` ( ( projh ` H ) ` A ) ) e. RR ) |
3 |
1 2
|
syl |
|- ( ( H e. CH /\ A e. ~H ) -> ( normh ` ( ( projh ` H ) ` A ) ) e. RR ) |
4 |
|
normcl |
|- ( A e. ~H -> ( normh ` A ) e. RR ) |
5 |
4
|
adantl |
|- ( ( H e. CH /\ A e. ~H ) -> ( normh ` A ) e. RR ) |
6 |
3 5
|
eqleltd |
|- ( ( H e. CH /\ A e. ~H ) -> ( ( normh ` ( ( projh ` H ) ` A ) ) = ( normh ` A ) <-> ( ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) /\ -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) ) ) |
7 |
|
pjnorm |
|- ( ( H e. CH /\ A e. ~H ) -> ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) ) |
8 |
7
|
biantrurd |
|- ( ( H e. CH /\ A e. ~H ) -> ( -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) <-> ( ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) /\ -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) ) ) |
9 |
|
pjnel |
|- ( ( H e. CH /\ A e. ~H ) -> ( -. A e. H <-> ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) ) |
10 |
9
|
con1bid |
|- ( ( H e. CH /\ A e. ~H ) -> ( -. ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) <-> A e. H ) ) |
11 |
6 8 10
|
3bitr2rd |
|- ( ( H e. CH /\ A e. ~H ) -> ( A e. H <-> ( normh ` ( ( projh ` H ) ` A ) ) = ( normh ` A ) ) ) |