Step |
Hyp |
Ref |
Expression |
1 |
|
norm1ex.1 |
|- H e. SH |
2 |
|
neeq1 |
|- ( x = z -> ( x =/= 0h <-> z =/= 0h ) ) |
3 |
2
|
cbvrexvw |
|- ( E. x e. H x =/= 0h <-> E. z e. H z =/= 0h ) |
4 |
1
|
sheli |
|- ( z e. H -> z e. ~H ) |
5 |
|
normcl |
|- ( z e. ~H -> ( normh ` z ) e. RR ) |
6 |
4 5
|
syl |
|- ( z e. H -> ( normh ` z ) e. RR ) |
7 |
6
|
adantr |
|- ( ( z e. H /\ z =/= 0h ) -> ( normh ` z ) e. RR ) |
8 |
|
normne0 |
|- ( z e. ~H -> ( ( normh ` z ) =/= 0 <-> z =/= 0h ) ) |
9 |
4 8
|
syl |
|- ( z e. H -> ( ( normh ` z ) =/= 0 <-> z =/= 0h ) ) |
10 |
9
|
biimpar |
|- ( ( z e. H /\ z =/= 0h ) -> ( normh ` z ) =/= 0 ) |
11 |
7 10
|
rereccld |
|- ( ( z e. H /\ z =/= 0h ) -> ( 1 / ( normh ` z ) ) e. RR ) |
12 |
11
|
recnd |
|- ( ( z e. H /\ z =/= 0h ) -> ( 1 / ( normh ` z ) ) e. CC ) |
13 |
|
simpl |
|- ( ( z e. H /\ z =/= 0h ) -> z e. H ) |
14 |
|
shmulcl |
|- ( ( H e. SH /\ ( 1 / ( normh ` z ) ) e. CC /\ z e. H ) -> ( ( 1 / ( normh ` z ) ) .h z ) e. H ) |
15 |
1 14
|
mp3an1 |
|- ( ( ( 1 / ( normh ` z ) ) e. CC /\ z e. H ) -> ( ( 1 / ( normh ` z ) ) .h z ) e. H ) |
16 |
12 13 15
|
syl2anc |
|- ( ( z e. H /\ z =/= 0h ) -> ( ( 1 / ( normh ` z ) ) .h z ) e. H ) |
17 |
|
norm1 |
|- ( ( z e. ~H /\ z =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 ) |
18 |
4 17
|
sylan |
|- ( ( z e. H /\ z =/= 0h ) -> ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 ) |
19 |
|
fveqeq2 |
|- ( y = ( ( 1 / ( normh ` z ) ) .h z ) -> ( ( normh ` y ) = 1 <-> ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 ) ) |
20 |
19
|
rspcev |
|- ( ( ( ( 1 / ( normh ` z ) ) .h z ) e. H /\ ( normh ` ( ( 1 / ( normh ` z ) ) .h z ) ) = 1 ) -> E. y e. H ( normh ` y ) = 1 ) |
21 |
16 18 20
|
syl2anc |
|- ( ( z e. H /\ z =/= 0h ) -> E. y e. H ( normh ` y ) = 1 ) |
22 |
21
|
rexlimiva |
|- ( E. z e. H z =/= 0h -> E. y e. H ( normh ` y ) = 1 ) |
23 |
|
ax-1ne0 |
|- 1 =/= 0 |
24 |
23
|
neii |
|- -. 1 = 0 |
25 |
|
eqeq1 |
|- ( ( normh ` y ) = 1 -> ( ( normh ` y ) = 0 <-> 1 = 0 ) ) |
26 |
24 25
|
mtbiri |
|- ( ( normh ` y ) = 1 -> -. ( normh ` y ) = 0 ) |
27 |
1
|
sheli |
|- ( y e. H -> y e. ~H ) |
28 |
|
norm-i |
|- ( y e. ~H -> ( ( normh ` y ) = 0 <-> y = 0h ) ) |
29 |
27 28
|
syl |
|- ( y e. H -> ( ( normh ` y ) = 0 <-> y = 0h ) ) |
30 |
29
|
necon3bbid |
|- ( y e. H -> ( -. ( normh ` y ) = 0 <-> y =/= 0h ) ) |
31 |
26 30
|
syl5ib |
|- ( y e. H -> ( ( normh ` y ) = 1 -> y =/= 0h ) ) |
32 |
31
|
reximia |
|- ( E. y e. H ( normh ` y ) = 1 -> E. y e. H y =/= 0h ) |
33 |
|
neeq1 |
|- ( y = z -> ( y =/= 0h <-> z =/= 0h ) ) |
34 |
33
|
cbvrexvw |
|- ( E. y e. H y =/= 0h <-> E. z e. H z =/= 0h ) |
35 |
32 34
|
sylib |
|- ( E. y e. H ( normh ` y ) = 1 -> E. z e. H z =/= 0h ) |
36 |
22 35
|
impbii |
|- ( E. z e. H z =/= 0h <-> E. y e. H ( normh ` y ) = 1 ) |
37 |
3 36
|
bitri |
|- ( E. x e. H x =/= 0h <-> E. y e. H ( normh ` y ) = 1 ) |