Step |
Hyp |
Ref |
Expression |
1 |
|
pjf.k |
|- K = ( proj ` W ) |
2 |
|
eqid |
|- ( LSubSp ` W ) = ( LSubSp ` W ) |
3 |
|
eqid |
|- ( LSSum ` W ) = ( LSSum ` W ) |
4 |
|
eqid |
|- ( 0g ` W ) = ( 0g ` W ) |
5 |
|
eqid |
|- ( proj1 ` W ) = ( proj1 ` W ) |
6 |
|
phllmod |
|- ( W e. PreHil -> W e. LMod ) |
7 |
6
|
adantr |
|- ( ( W e. PreHil /\ x e. dom K ) -> W e. LMod ) |
8 |
|
eqid |
|- ( Base ` W ) = ( Base ` W ) |
9 |
|
eqid |
|- ( ocv ` W ) = ( ocv ` W ) |
10 |
8 2 9 3 1
|
pjdm2 |
|- ( W e. PreHil -> ( x e. dom K <-> ( x e. ( LSubSp ` W ) /\ ( x ( LSSum ` W ) ( ( ocv ` W ) ` x ) ) = ( Base ` W ) ) ) ) |
11 |
10
|
simprbda |
|- ( ( W e. PreHil /\ x e. dom K ) -> x e. ( LSubSp ` W ) ) |
12 |
8 2
|
lssss |
|- ( x e. ( LSubSp ` W ) -> x C_ ( Base ` W ) ) |
13 |
11 12
|
syl |
|- ( ( W e. PreHil /\ x e. dom K ) -> x C_ ( Base ` W ) ) |
14 |
8 9 2
|
ocvlss |
|- ( ( W e. PreHil /\ x C_ ( Base ` W ) ) -> ( ( ocv ` W ) ` x ) e. ( LSubSp ` W ) ) |
15 |
13 14
|
syldan |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( ( ocv ` W ) ` x ) e. ( LSubSp ` W ) ) |
16 |
9 2 4
|
ocvin |
|- ( ( W e. PreHil /\ x e. ( LSubSp ` W ) ) -> ( x i^i ( ( ocv ` W ) ` x ) ) = { ( 0g ` W ) } ) |
17 |
11 16
|
syldan |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( x i^i ( ( ocv ` W ) ` x ) ) = { ( 0g ` W ) } ) |
18 |
2 3 4 5 7 11 15 17
|
pj1lmhm |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( x ( proj1 ` W ) ( ( ocv ` W ) ` x ) ) e. ( ( W |`s ( x ( LSSum ` W ) ( ( ocv ` W ) ` x ) ) ) LMHom W ) ) |
19 |
10
|
simplbda |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( x ( LSSum ` W ) ( ( ocv ` W ) ` x ) ) = ( Base ` W ) ) |
20 |
19
|
oveq2d |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( W |`s ( x ( LSSum ` W ) ( ( ocv ` W ) ` x ) ) ) = ( W |`s ( Base ` W ) ) ) |
21 |
8
|
ressid |
|- ( W e. PreHil -> ( W |`s ( Base ` W ) ) = W ) |
22 |
21
|
adantr |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( W |`s ( Base ` W ) ) = W ) |
23 |
20 22
|
eqtrd |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( W |`s ( x ( LSSum ` W ) ( ( ocv ` W ) ` x ) ) ) = W ) |
24 |
23
|
oveq1d |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( ( W |`s ( x ( LSSum ` W ) ( ( ocv ` W ) ` x ) ) ) LMHom W ) = ( W LMHom W ) ) |
25 |
18 24
|
eleqtrd |
|- ( ( W e. PreHil /\ x e. dom K ) -> ( x ( proj1 ` W ) ( ( ocv ` W ) ` x ) ) e. ( W LMHom W ) ) |
26 |
9 5 1
|
pjfval2 |
|- K = ( x e. dom K |-> ( x ( proj1 ` W ) ( ( ocv ` W ) ` x ) ) ) |
27 |
25 26
|
fmptd |
|- ( W e. PreHil -> K : dom K --> ( W LMHom W ) ) |