Step |
Hyp |
Ref |
Expression |
1 |
|
ip2eq.h |
|- ., = ( .i ` W ) |
2 |
|
ip2eq.v |
|- V = ( Base ` W ) |
3 |
|
oveq2 |
|- ( A = B -> ( x ., A ) = ( x ., B ) ) |
4 |
3
|
ralrimivw |
|- ( A = B -> A. x e. V ( x ., A ) = ( x ., B ) ) |
5 |
|
phllmod |
|- ( W e. PreHil -> W e. LMod ) |
6 |
|
eqid |
|- ( -g ` W ) = ( -g ` W ) |
7 |
2 6
|
lmodvsubcl |
|- ( ( W e. LMod /\ A e. V /\ B e. V ) -> ( A ( -g ` W ) B ) e. V ) |
8 |
5 7
|
syl3an1 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A ( -g ` W ) B ) e. V ) |
9 |
|
oveq1 |
|- ( x = ( A ( -g ` W ) B ) -> ( x ., A ) = ( ( A ( -g ` W ) B ) ., A ) ) |
10 |
|
oveq1 |
|- ( x = ( A ( -g ` W ) B ) -> ( x ., B ) = ( ( A ( -g ` W ) B ) ., B ) ) |
11 |
9 10
|
eqeq12d |
|- ( x = ( A ( -g ` W ) B ) -> ( ( x ., A ) = ( x ., B ) <-> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) ) |
12 |
11
|
rspcv |
|- ( ( A ( -g ` W ) B ) e. V -> ( A. x e. V ( x ., A ) = ( x ., B ) -> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) ) |
13 |
8 12
|
syl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A. x e. V ( x ., A ) = ( x ., B ) -> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) ) |
14 |
|
simp1 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> W e. PreHil ) |
15 |
|
simp2 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> A e. V ) |
16 |
|
simp3 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> B e. V ) |
17 |
|
eqid |
|- ( Scalar ` W ) = ( Scalar ` W ) |
18 |
|
eqid |
|- ( -g ` ( Scalar ` W ) ) = ( -g ` ( Scalar ` W ) ) |
19 |
17 1 2 6 18
|
ipsubdi |
|- ( ( W e. PreHil /\ ( ( A ( -g ` W ) B ) e. V /\ A e. V /\ B e. V ) ) -> ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` ( Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) ) |
20 |
14 8 15 16 19
|
syl13anc |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` ( Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) ) |
21 |
20
|
eqeq1d |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( 0g ` ( Scalar ` W ) ) <-> ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` ( Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` ( Scalar ` W ) ) ) ) |
22 |
|
eqid |
|- ( 0g ` ( Scalar ` W ) ) = ( 0g ` ( Scalar ` W ) ) |
23 |
|
eqid |
|- ( 0g ` W ) = ( 0g ` W ) |
24 |
17 1 2 22 23
|
ipeq0 |
|- ( ( W e. PreHil /\ ( A ( -g ` W ) B ) e. V ) -> ( ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( 0g ` ( Scalar ` W ) ) <-> ( A ( -g ` W ) B ) = ( 0g ` W ) ) ) |
25 |
14 8 24
|
syl2anc |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( A ( -g ` W ) B ) ., ( A ( -g ` W ) B ) ) = ( 0g ` ( Scalar ` W ) ) <-> ( A ( -g ` W ) B ) = ( 0g ` W ) ) ) |
26 |
21 25
|
bitr3d |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` ( Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` ( Scalar ` W ) ) <-> ( A ( -g ` W ) B ) = ( 0g ` W ) ) ) |
27 |
5
|
3ad2ant1 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> W e. LMod ) |
28 |
17
|
lmodfgrp |
|- ( W e. LMod -> ( Scalar ` W ) e. Grp ) |
29 |
27 28
|
syl |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( Scalar ` W ) e. Grp ) |
30 |
|
eqid |
|- ( Base ` ( Scalar ` W ) ) = ( Base ` ( Scalar ` W ) ) |
31 |
17 1 2 30
|
ipcl |
|- ( ( W e. PreHil /\ ( A ( -g ` W ) B ) e. V /\ A e. V ) -> ( ( A ( -g ` W ) B ) ., A ) e. ( Base ` ( Scalar ` W ) ) ) |
32 |
14 8 15 31
|
syl3anc |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., A ) e. ( Base ` ( Scalar ` W ) ) ) |
33 |
17 1 2 30
|
ipcl |
|- ( ( W e. PreHil /\ ( A ( -g ` W ) B ) e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., B ) e. ( Base ` ( Scalar ` W ) ) ) |
34 |
14 8 16 33
|
syl3anc |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) ., B ) e. ( Base ` ( Scalar ` W ) ) ) |
35 |
30 22 18
|
grpsubeq0 |
|- ( ( ( Scalar ` W ) e. Grp /\ ( ( A ( -g ` W ) B ) ., A ) e. ( Base ` ( Scalar ` W ) ) /\ ( ( A ( -g ` W ) B ) ., B ) e. ( Base ` ( Scalar ` W ) ) ) -> ( ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` ( Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` ( Scalar ` W ) ) <-> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) ) |
36 |
29 32 34 35
|
syl3anc |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( ( A ( -g ` W ) B ) ., A ) ( -g ` ( Scalar ` W ) ) ( ( A ( -g ` W ) B ) ., B ) ) = ( 0g ` ( Scalar ` W ) ) <-> ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) ) ) |
37 |
|
lmodgrp |
|- ( W e. LMod -> W e. Grp ) |
38 |
5 37
|
syl |
|- ( W e. PreHil -> W e. Grp ) |
39 |
2 23 6
|
grpsubeq0 |
|- ( ( W e. Grp /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) = ( 0g ` W ) <-> A = B ) ) |
40 |
38 39
|
syl3an1 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( A ( -g ` W ) B ) = ( 0g ` W ) <-> A = B ) ) |
41 |
26 36 40
|
3bitr3d |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( ( ( A ( -g ` W ) B ) ., A ) = ( ( A ( -g ` W ) B ) ., B ) <-> A = B ) ) |
42 |
13 41
|
sylibd |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A. x e. V ( x ., A ) = ( x ., B ) -> A = B ) ) |
43 |
4 42
|
impbid2 |
|- ( ( W e. PreHil /\ A e. V /\ B e. V ) -> ( A = B <-> A. x e. V ( x ., A ) = ( x ., B ) ) ) |