Step |
Hyp |
Ref |
Expression |
1 |
|
pjhth.1 |
|- H e. CH |
2 |
|
pjhth.2 |
|- ( ph -> A e. ~H ) |
3 |
2
|
adantr |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> A e. ~H ) |
4 |
1
|
cheli |
|- ( x e. H -> x e. ~H ) |
5 |
4
|
ad2antrl |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> x e. ~H ) |
6 |
|
hvsubcl |
|- ( ( A e. ~H /\ x e. ~H ) -> ( A -h x ) e. ~H ) |
7 |
3 5 6
|
syl2anc |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> ( A -h x ) e. ~H ) |
8 |
3
|
adantr |
|- ( ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) /\ y e. H ) -> A e. ~H ) |
9 |
|
simplrl |
|- ( ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) /\ y e. H ) -> x e. H ) |
10 |
|
simpr |
|- ( ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) /\ y e. H ) -> y e. H ) |
11 |
|
simplrr |
|- ( ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) /\ y e. H ) -> A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) |
12 |
|
eqid |
|- ( ( ( A -h x ) .ih y ) / ( ( y .ih y ) + 1 ) ) = ( ( ( A -h x ) .ih y ) / ( ( y .ih y ) + 1 ) ) |
13 |
1 8 9 10 11 12
|
pjhthlem1 |
|- ( ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) /\ y e. H ) -> ( ( A -h x ) .ih y ) = 0 ) |
14 |
13
|
ralrimiva |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> A. y e. H ( ( A -h x ) .ih y ) = 0 ) |
15 |
1
|
chshii |
|- H e. SH |
16 |
|
shocel |
|- ( H e. SH -> ( ( A -h x ) e. ( _|_ ` H ) <-> ( ( A -h x ) e. ~H /\ A. y e. H ( ( A -h x ) .ih y ) = 0 ) ) ) |
17 |
15 16
|
ax-mp |
|- ( ( A -h x ) e. ( _|_ ` H ) <-> ( ( A -h x ) e. ~H /\ A. y e. H ( ( A -h x ) .ih y ) = 0 ) ) |
18 |
7 14 17
|
sylanbrc |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> ( A -h x ) e. ( _|_ ` H ) ) |
19 |
|
hvpncan3 |
|- ( ( x e. ~H /\ A e. ~H ) -> ( x +h ( A -h x ) ) = A ) |
20 |
5 3 19
|
syl2anc |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> ( x +h ( A -h x ) ) = A ) |
21 |
20
|
eqcomd |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> A = ( x +h ( A -h x ) ) ) |
22 |
|
oveq2 |
|- ( y = ( A -h x ) -> ( x +h y ) = ( x +h ( A -h x ) ) ) |
23 |
22
|
rspceeqv |
|- ( ( ( A -h x ) e. ( _|_ ` H ) /\ A = ( x +h ( A -h x ) ) ) -> E. y e. ( _|_ ` H ) A = ( x +h y ) ) |
24 |
18 21 23
|
syl2anc |
|- ( ( ph /\ ( x e. H /\ A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) ) -> E. y e. ( _|_ ` H ) A = ( x +h y ) ) |
25 |
|
df-hba |
|- ~H = ( BaseSet ` <. <. +h , .h >. , normh >. ) |
26 |
|
eqid |
|- <. <. +h , .h >. , normh >. = <. <. +h , .h >. , normh >. |
27 |
26
|
hhvs |
|- -h = ( -v ` <. <. +h , .h >. , normh >. ) |
28 |
26
|
hhnm |
|- normh = ( normCV ` <. <. +h , .h >. , normh >. ) |
29 |
|
eqid |
|- <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. = <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. |
30 |
29 15
|
hhssba |
|- H = ( BaseSet ` <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. ) |
31 |
26
|
hhph |
|- <. <. +h , .h >. , normh >. e. CPreHilOLD |
32 |
31
|
a1i |
|- ( ph -> <. <. +h , .h >. , normh >. e. CPreHilOLD ) |
33 |
26 29
|
hhsst |
|- ( H e. SH -> <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. ( SubSp ` <. <. +h , .h >. , normh >. ) ) |
34 |
15 33
|
ax-mp |
|- <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. ( SubSp ` <. <. +h , .h >. , normh >. ) |
35 |
29 1
|
hhssbnOLD |
|- <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. CBan |
36 |
|
elin |
|- ( <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. ( ( SubSp ` <. <. +h , .h >. , normh >. ) i^i CBan ) <-> ( <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. ( SubSp ` <. <. +h , .h >. , normh >. ) /\ <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. CBan ) ) |
37 |
34 35 36
|
mpbir2an |
|- <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. ( ( SubSp ` <. <. +h , .h >. , normh >. ) i^i CBan ) |
38 |
37
|
a1i |
|- ( ph -> <. <. ( +h |` ( H X. H ) ) , ( .h |` ( CC X. H ) ) >. , ( normh |` H ) >. e. ( ( SubSp ` <. <. +h , .h >. , normh >. ) i^i CBan ) ) |
39 |
25 27 28 30 32 38 2
|
minveco |
|- ( ph -> E! x e. H A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) |
40 |
|
reurex |
|- ( E! x e. H A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) -> E. x e. H A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) |
41 |
39 40
|
syl |
|- ( ph -> E. x e. H A. z e. H ( normh ` ( A -h x ) ) <_ ( normh ` ( A -h z ) ) ) |
42 |
24 41
|
reximddv |
|- ( ph -> E. x e. H E. y e. ( _|_ ` H ) A = ( x +h y ) ) |