Step |
Hyp |
Ref |
Expression |
1 |
|
pjnorm.1 |
|- H e. CH |
2 |
|
pjnorm.2 |
|- A e. ~H |
3 |
1 2
|
pjnormi |
|- ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) |
4 |
3
|
biantrur |
|- ( ( normh ` A ) =/= ( normh ` ( ( projh ` H ) ` A ) ) <-> ( ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) /\ ( normh ` A ) =/= ( normh ` ( ( projh ` H ) ` A ) ) ) ) |
5 |
1 2
|
pjoc1i |
|- ( A e. H <-> ( ( projh ` ( _|_ ` H ) ) ` A ) = 0h ) |
6 |
1 2
|
pjpythi |
|- ( ( normh ` A ) ^ 2 ) = ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) ) |
7 |
|
sq0 |
|- ( 0 ^ 2 ) = 0 |
8 |
7
|
oveq2i |
|- ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( 0 ^ 2 ) ) = ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + 0 ) |
9 |
1 2
|
pjhclii |
|- ( ( projh ` H ) ` A ) e. ~H |
10 |
9
|
normcli |
|- ( normh ` ( ( projh ` H ) ` A ) ) e. RR |
11 |
10
|
resqcli |
|- ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) e. RR |
12 |
11
|
recni |
|- ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) e. CC |
13 |
12
|
addid1i |
|- ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + 0 ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) |
14 |
8 13
|
eqtr2i |
|- ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) = ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( 0 ^ 2 ) ) |
15 |
6 14
|
eqeq12i |
|- ( ( ( normh ` A ) ^ 2 ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) <-> ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) ) = ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( 0 ^ 2 ) ) ) |
16 |
1
|
choccli |
|- ( _|_ ` H ) e. CH |
17 |
16 2
|
pjhclii |
|- ( ( projh ` ( _|_ ` H ) ) ` A ) e. ~H |
18 |
17
|
normcli |
|- ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) e. RR |
19 |
18
|
resqcli |
|- ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) e. RR |
20 |
19
|
recni |
|- ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) e. CC |
21 |
|
0cn |
|- 0 e. CC |
22 |
21
|
sqcli |
|- ( 0 ^ 2 ) e. CC |
23 |
12 20 22
|
addcani |
|- ( ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) ) = ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( 0 ^ 2 ) ) <-> ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) = ( 0 ^ 2 ) ) |
24 |
|
normge0 |
|- ( ( ( projh ` ( _|_ ` H ) ) ` A ) e. ~H -> 0 <_ ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ) |
25 |
17 24
|
ax-mp |
|- 0 <_ ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) |
26 |
|
0le0 |
|- 0 <_ 0 |
27 |
|
0re |
|- 0 e. RR |
28 |
18 27
|
sq11i |
|- ( ( 0 <_ ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) /\ 0 <_ 0 ) -> ( ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) = ( 0 ^ 2 ) <-> ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) = 0 ) ) |
29 |
25 26 28
|
mp2an |
|- ( ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) = ( 0 ^ 2 ) <-> ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) = 0 ) |
30 |
17
|
norm-i-i |
|- ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) = 0 <-> ( ( projh ` ( _|_ ` H ) ) ` A ) = 0h ) |
31 |
23 29 30
|
3bitri |
|- ( ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( ( normh ` ( ( projh ` ( _|_ ` H ) ) ` A ) ) ^ 2 ) ) = ( ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) + ( 0 ^ 2 ) ) <-> ( ( projh ` ( _|_ ` H ) ) ` A ) = 0h ) |
32 |
15 31
|
bitr2i |
|- ( ( ( projh ` ( _|_ ` H ) ) ` A ) = 0h <-> ( ( normh ` A ) ^ 2 ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) ) |
33 |
|
normge0 |
|- ( A e. ~H -> 0 <_ ( normh ` A ) ) |
34 |
2 33
|
ax-mp |
|- 0 <_ ( normh ` A ) |
35 |
|
normge0 |
|- ( ( ( projh ` H ) ` A ) e. ~H -> 0 <_ ( normh ` ( ( projh ` H ) ` A ) ) ) |
36 |
9 35
|
ax-mp |
|- 0 <_ ( normh ` ( ( projh ` H ) ` A ) ) |
37 |
2
|
normcli |
|- ( normh ` A ) e. RR |
38 |
37 10
|
sq11i |
|- ( ( 0 <_ ( normh ` A ) /\ 0 <_ ( normh ` ( ( projh ` H ) ` A ) ) ) -> ( ( ( normh ` A ) ^ 2 ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) <-> ( normh ` A ) = ( normh ` ( ( projh ` H ) ` A ) ) ) ) |
39 |
34 36 38
|
mp2an |
|- ( ( ( normh ` A ) ^ 2 ) = ( ( normh ` ( ( projh ` H ) ` A ) ) ^ 2 ) <-> ( normh ` A ) = ( normh ` ( ( projh ` H ) ` A ) ) ) |
40 |
5 32 39
|
3bitri |
|- ( A e. H <-> ( normh ` A ) = ( normh ` ( ( projh ` H ) ` A ) ) ) |
41 |
40
|
necon3bbii |
|- ( -. A e. H <-> ( normh ` A ) =/= ( normh ` ( ( projh ` H ) ` A ) ) ) |
42 |
10 37
|
ltleni |
|- ( ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) <-> ( ( normh ` ( ( projh ` H ) ` A ) ) <_ ( normh ` A ) /\ ( normh ` A ) =/= ( normh ` ( ( projh ` H ) ` A ) ) ) ) |
43 |
4 41 42
|
3bitr4i |
|- ( -. A e. H <-> ( normh ` ( ( projh ` H ) ` A ) ) < ( normh ` A ) ) |