| Step |
Hyp |
Ref |
Expression |
| 1 |
|
hdmaprnlem15.h |
|- H = ( LHyp ` K ) |
| 2 |
|
hdmaprnlem15.u |
|- U = ( ( DVecH ` K ) ` W ) |
| 3 |
|
hdmaprnlem15.v |
|- V = ( Base ` U ) |
| 4 |
|
hdmaprnlem15.n |
|- N = ( LSpan ` U ) |
| 5 |
|
hdmaprnlem15.c |
|- C = ( ( LCDual ` K ) ` W ) |
| 6 |
|
hdmaprnlem15.d |
|- D = ( Base ` C ) |
| 7 |
|
hdmaprnlem15.q |
|- .0. = ( 0g ` C ) |
| 8 |
|
hdmaprnlem15.l |
|- L = ( LSpan ` C ) |
| 9 |
|
hdmaprnlem15.m |
|- M = ( ( mapd ` K ) ` W ) |
| 10 |
|
hdmaprnlem15.s |
|- S = ( ( HDMap ` K ) ` W ) |
| 11 |
|
hdmaprnlem15.k |
|- ( ph -> ( K e. HL /\ W e. H ) ) |
| 12 |
|
hdmaprnlem17.se |
|- ( ph -> s e. D ) |
| 13 |
|
eleq1 |
|- ( s = .0. -> ( s e. ran S <-> .0. e. ran S ) ) |
| 14 |
11
|
adantr |
|- ( ( ph /\ s =/= .0. ) -> ( K e. HL /\ W e. H ) ) |
| 15 |
12
|
anim1i |
|- ( ( ph /\ s =/= .0. ) -> ( s e. D /\ s =/= .0. ) ) |
| 16 |
|
eldifsn |
|- ( s e. ( D \ { .0. } ) <-> ( s e. D /\ s =/= .0. ) ) |
| 17 |
15 16
|
sylibr |
|- ( ( ph /\ s =/= .0. ) -> s e. ( D \ { .0. } ) ) |
| 18 |
1 2 3 4 5 6 7 8 9 10 14 17
|
hdmaprnlem16N |
|- ( ( ph /\ s =/= .0. ) -> s e. ran S ) |
| 19 |
|
eqid |
|- ( 0g ` U ) = ( 0g ` U ) |
| 20 |
1 2 19 5 7 10 11
|
hdmapval0 |
|- ( ph -> ( S ` ( 0g ` U ) ) = .0. ) |
| 21 |
1 2 3 10 11
|
hdmapfnN |
|- ( ph -> S Fn V ) |
| 22 |
1 2 11
|
dvhlmod |
|- ( ph -> U e. LMod ) |
| 23 |
3 19
|
lmod0vcl |
|- ( U e. LMod -> ( 0g ` U ) e. V ) |
| 24 |
22 23
|
syl |
|- ( ph -> ( 0g ` U ) e. V ) |
| 25 |
|
fnfvelrn |
|- ( ( S Fn V /\ ( 0g ` U ) e. V ) -> ( S ` ( 0g ` U ) ) e. ran S ) |
| 26 |
21 24 25
|
syl2anc |
|- ( ph -> ( S ` ( 0g ` U ) ) e. ran S ) |
| 27 |
20 26
|
eqeltrrd |
|- ( ph -> .0. e. ran S ) |
| 28 |
13 18 27
|
pm2.61ne |
|- ( ph -> s e. ran S ) |