Step |
Hyp |
Ref |
Expression |
1 |
|
phlsrng.f |
|- F = ( Scalar ` W ) |
2 |
|
phllmhm.h |
|- ., = ( .i ` W ) |
3 |
|
phllmhm.v |
|- V = ( Base ` W ) |
4 |
|
ip0l.z |
|- Z = ( 0g ` F ) |
5 |
|
ip0l.o |
|- .0. = ( 0g ` W ) |
6 |
|
eqid |
|- ( *r ` F ) = ( *r ` F ) |
7 |
3 1 2 5 6 4
|
isphl |
|- ( W e. PreHil <-> ( W e. LVec /\ F e. *Ring /\ A. x e. V ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom ( ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) ) ) |
8 |
7
|
simp3bi |
|- ( W e. PreHil -> A. x e. V ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom ( ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) ) |
9 |
|
simp2 |
|- ( ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom ( ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) -> ( ( x ., x ) = Z -> x = .0. ) ) |
10 |
9
|
ralimi |
|- ( A. x e. V ( ( y e. V |-> ( y ., x ) ) e. ( W LMHom ( ringLMod ` F ) ) /\ ( ( x ., x ) = Z -> x = .0. ) /\ A. y e. V ( ( *r ` F ) ` ( x ., y ) ) = ( y ., x ) ) -> A. x e. V ( ( x ., x ) = Z -> x = .0. ) ) |
11 |
8 10
|
syl |
|- ( W e. PreHil -> A. x e. V ( ( x ., x ) = Z -> x = .0. ) ) |
12 |
|
oveq12 |
|- ( ( x = A /\ x = A ) -> ( x ., x ) = ( A ., A ) ) |
13 |
12
|
anidms |
|- ( x = A -> ( x ., x ) = ( A ., A ) ) |
14 |
13
|
eqeq1d |
|- ( x = A -> ( ( x ., x ) = Z <-> ( A ., A ) = Z ) ) |
15 |
|
eqeq1 |
|- ( x = A -> ( x = .0. <-> A = .0. ) ) |
16 |
14 15
|
imbi12d |
|- ( x = A -> ( ( ( x ., x ) = Z -> x = .0. ) <-> ( ( A ., A ) = Z -> A = .0. ) ) ) |
17 |
16
|
rspccva |
|- ( ( A. x e. V ( ( x ., x ) = Z -> x = .0. ) /\ A e. V ) -> ( ( A ., A ) = Z -> A = .0. ) ) |
18 |
11 17
|
sylan |
|- ( ( W e. PreHil /\ A e. V ) -> ( ( A ., A ) = Z -> A = .0. ) ) |
19 |
1 2 3 4 5
|
ip0l |
|- ( ( W e. PreHil /\ A e. V ) -> ( .0. ., A ) = Z ) |
20 |
|
oveq1 |
|- ( A = .0. -> ( A ., A ) = ( .0. ., A ) ) |
21 |
20
|
eqeq1d |
|- ( A = .0. -> ( ( A ., A ) = Z <-> ( .0. ., A ) = Z ) ) |
22 |
19 21
|
syl5ibrcom |
|- ( ( W e. PreHil /\ A e. V ) -> ( A = .0. -> ( A ., A ) = Z ) ) |
23 |
18 22
|
impbid |
|- ( ( W e. PreHil /\ A e. V ) -> ( ( A ., A ) = Z <-> A = .0. ) ) |