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 |
1
|
phlsrng |
|- ( W e. PreHil -> F e. *Ring ) |
6 |
5
|
3ad2ant1 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> F e. *Ring ) |
7 |
|
eqid |
|- ( *rf ` F ) = ( *rf ` F ) |
8 |
|
eqid |
|- ( Base ` F ) = ( Base ` F ) |
9 |
7 8
|
srngf1o |
|- ( F e. *Ring -> ( *rf ` F ) : ( Base ` F ) -1-1-onto-> ( Base ` F ) ) |
10 |
|
f1of1 |
|- ( ( *rf ` F ) : ( Base ` F ) -1-1-onto-> ( Base ` F ) -> ( *rf ` F ) : ( Base ` F ) -1-1-> ( Base ` F ) ) |
11 |
6 9 10
|
3syl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( *rf ` F ) : ( Base ` F ) -1-1-> ( Base ` F ) ) |
12 |
1 2 3 8
|
ipcl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A ., B ) e. ( Base ` F ) ) |
13 |
|
phllmod |
|- ( W e. PreHil -> W e. LMod ) |
14 |
13
|
3ad2ant1 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> W e. LMod ) |
15 |
1 8 4
|
lmod0cl |
|- ( W e. LMod -> Z e. ( Base ` F ) ) |
16 |
14 15
|
syl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> Z e. ( Base ` F ) ) |
17 |
|
f1fveq |
|- ( ( ( *rf ` F ) : ( Base ` F ) -1-1-> ( Base ` F ) /\ ( ( A ., B ) e. ( Base ` F ) /\ Z e. ( Base ` F ) ) ) -> ( ( ( *rf ` F ) ` ( A ., B ) ) = ( ( *rf ` F ) ` Z ) <-> ( A ., B ) = Z ) ) |
18 |
11 12 16 17
|
syl12anc |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( *rf ` F ) ` ( A ., B ) ) = ( ( *rf ` F ) ` Z ) <-> ( A ., B ) = Z ) ) |
19 |
|
eqid |
|- ( *r ` F ) = ( *r ` F ) |
20 |
8 19 7
|
stafval |
|- ( ( A ., B ) e. ( Base ` F ) -> ( ( *rf ` F ) ` ( A ., B ) ) = ( ( *r ` F ) ` ( A ., B ) ) ) |
21 |
12 20
|
syl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( *rf ` F ) ` ( A ., B ) ) = ( ( *r ` F ) ` ( A ., B ) ) ) |
22 |
1 2 3 19
|
ipcj |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( *r ` F ) ` ( A ., B ) ) = ( B ., A ) ) |
23 |
21 22
|
eqtrd |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( *rf ` F ) ` ( A ., B ) ) = ( B ., A ) ) |
24 |
8 19 7
|
stafval |
|- ( Z e. ( Base ` F ) -> ( ( *rf ` F ) ` Z ) = ( ( *r ` F ) ` Z ) ) |
25 |
16 24
|
syl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( *rf ` F ) ` Z ) = ( ( *r ` F ) ` Z ) ) |
26 |
19 4
|
srng0 |
|- ( F e. *Ring -> ( ( *r ` F ) ` Z ) = Z ) |
27 |
6 26
|
syl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( *r ` F ) ` Z ) = Z ) |
28 |
25 27
|
eqtrd |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( *rf ` F ) ` Z ) = Z ) |
29 |
23 28
|
eqeq12d |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( *rf ` F ) ` ( A ., B ) ) = ( ( *rf ` F ) ` Z ) <-> ( B ., A ) = Z ) ) |
30 |
18 29
|
bitr3d |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ., B ) = Z <-> ( B ., A ) = Z ) ) |