Step |
Hyp |
Ref |
Expression |
1 |
|
nlelch.1 |
|- T e. LinFn |
2 |
|
nlelch.2 |
|- T e. ContFn |
3 |
1 2
|
riesz3i |
|- E. w e. ~H A. v e. ~H ( T ` v ) = ( v .ih w ) |
4 |
|
r19.26 |
|- ( A. v e. ~H ( ( T ` v ) = ( v .ih w ) /\ ( T ` v ) = ( v .ih u ) ) <-> ( A. v e. ~H ( T ` v ) = ( v .ih w ) /\ A. v e. ~H ( T ` v ) = ( v .ih u ) ) ) |
5 |
|
oveq12 |
|- ( ( ( T ` v ) = ( v .ih w ) /\ ( T ` v ) = ( v .ih u ) ) -> ( ( T ` v ) - ( T ` v ) ) = ( ( v .ih w ) - ( v .ih u ) ) ) |
6 |
5
|
adantl |
|- ( ( v e. ~H /\ ( ( T ` v ) = ( v .ih w ) /\ ( T ` v ) = ( v .ih u ) ) ) -> ( ( T ` v ) - ( T ` v ) ) = ( ( v .ih w ) - ( v .ih u ) ) ) |
7 |
1
|
lnfnfi |
|- T : ~H --> CC |
8 |
7
|
ffvelrni |
|- ( v e. ~H -> ( T ` v ) e. CC ) |
9 |
8
|
subidd |
|- ( v e. ~H -> ( ( T ` v ) - ( T ` v ) ) = 0 ) |
10 |
9
|
adantr |
|- ( ( v e. ~H /\ ( ( T ` v ) = ( v .ih w ) /\ ( T ` v ) = ( v .ih u ) ) ) -> ( ( T ` v ) - ( T ` v ) ) = 0 ) |
11 |
6 10
|
eqtr3d |
|- ( ( v e. ~H /\ ( ( T ` v ) = ( v .ih w ) /\ ( T ` v ) = ( v .ih u ) ) ) -> ( ( v .ih w ) - ( v .ih u ) ) = 0 ) |
12 |
11
|
ralimiaa |
|- ( A. v e. ~H ( ( T ` v ) = ( v .ih w ) /\ ( T ` v ) = ( v .ih u ) ) -> A. v e. ~H ( ( v .ih w ) - ( v .ih u ) ) = 0 ) |
13 |
4 12
|
sylbir |
|- ( ( A. v e. ~H ( T ` v ) = ( v .ih w ) /\ A. v e. ~H ( T ` v ) = ( v .ih u ) ) -> A. v e. ~H ( ( v .ih w ) - ( v .ih u ) ) = 0 ) |
14 |
|
hvsubcl |
|- ( ( w e. ~H /\ u e. ~H ) -> ( w -h u ) e. ~H ) |
15 |
|
oveq1 |
|- ( v = ( w -h u ) -> ( v .ih w ) = ( ( w -h u ) .ih w ) ) |
16 |
|
oveq1 |
|- ( v = ( w -h u ) -> ( v .ih u ) = ( ( w -h u ) .ih u ) ) |
17 |
15 16
|
oveq12d |
|- ( v = ( w -h u ) -> ( ( v .ih w ) - ( v .ih u ) ) = ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) ) |
18 |
17
|
eqeq1d |
|- ( v = ( w -h u ) -> ( ( ( v .ih w ) - ( v .ih u ) ) = 0 <-> ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) = 0 ) ) |
19 |
18
|
rspcv |
|- ( ( w -h u ) e. ~H -> ( A. v e. ~H ( ( v .ih w ) - ( v .ih u ) ) = 0 -> ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) = 0 ) ) |
20 |
14 19
|
syl |
|- ( ( w e. ~H /\ u e. ~H ) -> ( A. v e. ~H ( ( v .ih w ) - ( v .ih u ) ) = 0 -> ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) = 0 ) ) |
21 |
|
normcl |
|- ( ( w -h u ) e. ~H -> ( normh ` ( w -h u ) ) e. RR ) |
22 |
21
|
recnd |
|- ( ( w -h u ) e. ~H -> ( normh ` ( w -h u ) ) e. CC ) |
23 |
|
sqeq0 |
|- ( ( normh ` ( w -h u ) ) e. CC -> ( ( ( normh ` ( w -h u ) ) ^ 2 ) = 0 <-> ( normh ` ( w -h u ) ) = 0 ) ) |
24 |
22 23
|
syl |
|- ( ( w -h u ) e. ~H -> ( ( ( normh ` ( w -h u ) ) ^ 2 ) = 0 <-> ( normh ` ( w -h u ) ) = 0 ) ) |
25 |
|
norm-i |
|- ( ( w -h u ) e. ~H -> ( ( normh ` ( w -h u ) ) = 0 <-> ( w -h u ) = 0h ) ) |
26 |
24 25
|
bitrd |
|- ( ( w -h u ) e. ~H -> ( ( ( normh ` ( w -h u ) ) ^ 2 ) = 0 <-> ( w -h u ) = 0h ) ) |
27 |
14 26
|
syl |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( ( normh ` ( w -h u ) ) ^ 2 ) = 0 <-> ( w -h u ) = 0h ) ) |
28 |
|
normsq |
|- ( ( w -h u ) e. ~H -> ( ( normh ` ( w -h u ) ) ^ 2 ) = ( ( w -h u ) .ih ( w -h u ) ) ) |
29 |
14 28
|
syl |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( normh ` ( w -h u ) ) ^ 2 ) = ( ( w -h u ) .ih ( w -h u ) ) ) |
30 |
|
simpl |
|- ( ( w e. ~H /\ u e. ~H ) -> w e. ~H ) |
31 |
|
simpr |
|- ( ( w e. ~H /\ u e. ~H ) -> u e. ~H ) |
32 |
|
his2sub2 |
|- ( ( ( w -h u ) e. ~H /\ w e. ~H /\ u e. ~H ) -> ( ( w -h u ) .ih ( w -h u ) ) = ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) ) |
33 |
14 30 31 32
|
syl3anc |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( w -h u ) .ih ( w -h u ) ) = ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) ) |
34 |
29 33
|
eqtrd |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( normh ` ( w -h u ) ) ^ 2 ) = ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) ) |
35 |
34
|
eqeq1d |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( ( normh ` ( w -h u ) ) ^ 2 ) = 0 <-> ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) = 0 ) ) |
36 |
|
hvsubeq0 |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( w -h u ) = 0h <-> w = u ) ) |
37 |
27 35 36
|
3bitr3d |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( ( ( w -h u ) .ih w ) - ( ( w -h u ) .ih u ) ) = 0 <-> w = u ) ) |
38 |
20 37
|
sylibd |
|- ( ( w e. ~H /\ u e. ~H ) -> ( A. v e. ~H ( ( v .ih w ) - ( v .ih u ) ) = 0 -> w = u ) ) |
39 |
13 38
|
syl5 |
|- ( ( w e. ~H /\ u e. ~H ) -> ( ( A. v e. ~H ( T ` v ) = ( v .ih w ) /\ A. v e. ~H ( T ` v ) = ( v .ih u ) ) -> w = u ) ) |
40 |
39
|
rgen2 |
|- A. w e. ~H A. u e. ~H ( ( A. v e. ~H ( T ` v ) = ( v .ih w ) /\ A. v e. ~H ( T ` v ) = ( v .ih u ) ) -> w = u ) |
41 |
|
oveq2 |
|- ( w = u -> ( v .ih w ) = ( v .ih u ) ) |
42 |
41
|
eqeq2d |
|- ( w = u -> ( ( T ` v ) = ( v .ih w ) <-> ( T ` v ) = ( v .ih u ) ) ) |
43 |
42
|
ralbidv |
|- ( w = u -> ( A. v e. ~H ( T ` v ) = ( v .ih w ) <-> A. v e. ~H ( T ` v ) = ( v .ih u ) ) ) |
44 |
43
|
reu4 |
|- ( E! w e. ~H A. v e. ~H ( T ` v ) = ( v .ih w ) <-> ( E. w e. ~H A. v e. ~H ( T ` v ) = ( v .ih w ) /\ A. w e. ~H A. u e. ~H ( ( A. v e. ~H ( T ` v ) = ( v .ih w ) /\ A. v e. ~H ( T ` v ) = ( v .ih u ) ) -> w = u ) ) ) |
45 |
3 40 44
|
mpbir2an |
|- E! w e. ~H A. v e. ~H ( T ` v ) = ( v .ih w ) |