Step |
Hyp |
Ref |
Expression |
1 |
|
hstrlem3.1 |
|- S = ( x e. CH |-> ( ( projh ` x ) ` u ) ) |
2 |
|
hstrlem3.2 |
|- ( ph <-> ( u e. ( A \ B ) /\ ( normh ` u ) = 1 ) ) |
3 |
|
hstrlem3.3 |
|- A e. CH |
4 |
|
hstrlem3.4 |
|- B e. CH |
5 |
1
|
hstrlem2 |
|- ( A e. CH -> ( S ` A ) = ( ( projh ` A ) ` u ) ) |
6 |
3 5
|
ax-mp |
|- ( S ` A ) = ( ( projh ` A ) ` u ) |
7 |
6
|
fveq2i |
|- ( normh ` ( S ` A ) ) = ( normh ` ( ( projh ` A ) ` u ) ) |
8 |
|
eldifi |
|- ( u e. ( A \ B ) -> u e. A ) |
9 |
|
pjid |
|- ( ( A e. CH /\ u e. A ) -> ( ( projh ` A ) ` u ) = u ) |
10 |
3 9
|
mpan |
|- ( u e. A -> ( ( projh ` A ) ` u ) = u ) |
11 |
10
|
fveq2d |
|- ( u e. A -> ( normh ` ( ( projh ` A ) ` u ) ) = ( normh ` u ) ) |
12 |
|
eqeq2 |
|- ( ( normh ` u ) = 1 -> ( ( normh ` ( ( projh ` A ) ` u ) ) = ( normh ` u ) <-> ( normh ` ( ( projh ` A ) ` u ) ) = 1 ) ) |
13 |
11 12
|
syl5ib |
|- ( ( normh ` u ) = 1 -> ( u e. A -> ( normh ` ( ( projh ` A ) ` u ) ) = 1 ) ) |
14 |
8 13
|
mpan9 |
|- ( ( u e. ( A \ B ) /\ ( normh ` u ) = 1 ) -> ( normh ` ( ( projh ` A ) ` u ) ) = 1 ) |
15 |
2 14
|
sylbi |
|- ( ph -> ( normh ` ( ( projh ` A ) ` u ) ) = 1 ) |
16 |
7 15
|
eqtrid |
|- ( ph -> ( normh ` ( S ` A ) ) = 1 ) |