Step |
Hyp |
Ref |
Expression |
1 |
|
nlmmul0or.v |
|- V = ( Base ` W ) |
2 |
|
nlmmul0or.s |
|- .x. = ( .s ` W ) |
3 |
|
nlmmul0or.z |
|- .0. = ( 0g ` W ) |
4 |
|
nlmmul0or.f |
|- F = ( Scalar ` W ) |
5 |
|
nlmmul0or.k |
|- K = ( Base ` F ) |
6 |
|
nlmmul0or.o |
|- O = ( 0g ` F ) |
7 |
4
|
nlmngp2 |
|- ( W e. NrmMod -> F e. NrmGrp ) |
8 |
7
|
3ad2ant1 |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> F e. NrmGrp ) |
9 |
|
simp2 |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> A e. K ) |
10 |
|
eqid |
|- ( norm ` F ) = ( norm ` F ) |
11 |
5 10
|
nmcl |
|- ( ( F e. NrmGrp /\ A e. K ) -> ( ( norm ` F ) ` A ) e. RR ) |
12 |
8 9 11
|
syl2anc |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( norm ` F ) ` A ) e. RR ) |
13 |
12
|
recnd |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( norm ` F ) ` A ) e. CC ) |
14 |
|
nlmngp |
|- ( W e. NrmMod -> W e. NrmGrp ) |
15 |
14
|
3ad2ant1 |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> W e. NrmGrp ) |
16 |
|
simp3 |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> B e. V ) |
17 |
|
eqid |
|- ( norm ` W ) = ( norm ` W ) |
18 |
1 17
|
nmcl |
|- ( ( W e. NrmGrp /\ B e. V ) -> ( ( norm ` W ) ` B ) e. RR ) |
19 |
15 16 18
|
syl2anc |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( norm ` W ) ` B ) e. RR ) |
20 |
19
|
recnd |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( norm ` W ) ` B ) e. CC ) |
21 |
13 20
|
mul0ord |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( ( ( norm ` F ) ` A ) x. ( ( norm ` W ) ` B ) ) = 0 <-> ( ( ( norm ` F ) ` A ) = 0 \/ ( ( norm ` W ) ` B ) = 0 ) ) ) |
22 |
1 17 2 4 5 10
|
nmvs |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( norm ` W ) ` ( A .x. B ) ) = ( ( ( norm ` F ) ` A ) x. ( ( norm ` W ) ` B ) ) ) |
23 |
22
|
eqeq1d |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( ( norm ` W ) ` ( A .x. B ) ) = 0 <-> ( ( ( norm ` F ) ` A ) x. ( ( norm ` W ) ` B ) ) = 0 ) ) |
24 |
|
nlmlmod |
|- ( W e. NrmMod -> W e. LMod ) |
25 |
1 4 2 5
|
lmodvscl |
|- ( ( W e. LMod /\ A e. K /\ B e. V ) -> ( A .x. B ) e. V ) |
26 |
24 25
|
syl3an1 |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( A .x. B ) e. V ) |
27 |
1 17 3
|
nmeq0 |
|- ( ( W e. NrmGrp /\ ( A .x. B ) e. V ) -> ( ( ( norm ` W ) ` ( A .x. B ) ) = 0 <-> ( A .x. B ) = .0. ) ) |
28 |
15 26 27
|
syl2anc |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( ( norm ` W ) ` ( A .x. B ) ) = 0 <-> ( A .x. B ) = .0. ) ) |
29 |
23 28
|
bitr3d |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( ( ( norm ` F ) ` A ) x. ( ( norm ` W ) ` B ) ) = 0 <-> ( A .x. B ) = .0. ) ) |
30 |
5 10 6
|
nmeq0 |
|- ( ( F e. NrmGrp /\ A e. K ) -> ( ( ( norm ` F ) ` A ) = 0 <-> A = O ) ) |
31 |
8 9 30
|
syl2anc |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( ( norm ` F ) ` A ) = 0 <-> A = O ) ) |
32 |
1 17 3
|
nmeq0 |
|- ( ( W e. NrmGrp /\ B e. V ) -> ( ( ( norm ` W ) ` B ) = 0 <-> B = .0. ) ) |
33 |
15 16 32
|
syl2anc |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( ( norm ` W ) ` B ) = 0 <-> B = .0. ) ) |
34 |
31 33
|
orbi12d |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( ( ( norm ` F ) ` A ) = 0 \/ ( ( norm ` W ) ` B ) = 0 ) <-> ( A = O \/ B = .0. ) ) ) |
35 |
21 29 34
|
3bitr3d |
|- ( ( W e. NrmMod /\ A e. K /\ B e. V ) -> ( ( A .x. B ) = .0. <-> ( A = O \/ B = .0. ) ) ) |