Metamath Proof Explorer

Theorem mapdn0

Description: Transfer nonzero property from domain to range of projectivity mapd . (Contributed by NM, 12-Apr-2015)

Ref Expression
Hypotheses mapdindp.h 
|- H = ( LHyp ` K )
mapdindp.m 
|- M = ( ( mapd ` K ) ` W )
mapdindp.u 
|- U = ( ( DVecH ` K ) ` W )
mapdindp.v 
|- V = ( Base ` U )
mapdindp.n 
|- N = ( LSpan ` U )
mapdindp.c 
|- C = ( ( LCDual ` K ) ` W )
mapdindp.d 
|- D = ( Base ` C )
mapdindp.j 
|- J = ( LSpan ` C )
mapdindp.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdindp.f 
|- ( ph -> F e. D )
mapdindp.mx 
|- ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
mapdn0.o 
|- .0. = ( 0g ` U )
mapdn0.z 
|- Z = ( 0g ` C )
mapdn0.x 
|- ( ph -> X e. ( V \ { .0. } ) )
Assertion mapdn0 
|- ( ph -> F e. ( D \ { Z } ) )

Proof

Step Hyp Ref Expression
1 mapdindp.h 
 |-  H = ( LHyp ` K )
2 mapdindp.m 
 |-  M = ( ( mapd ` K ) ` W )
3 mapdindp.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 mapdindp.v 
 |-  V = ( Base ` U )
5 mapdindp.n 
 |-  N = ( LSpan ` U )
6 mapdindp.c 
 |-  C = ( ( LCDual ` K ) ` W )
7 mapdindp.d 
 |-  D = ( Base ` C )
8 mapdindp.j 
 |-  J = ( LSpan ` C )
9 mapdindp.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 mapdindp.f 
 |-  ( ph -> F e. D )
11 mapdindp.mx 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = ( J ` { F } ) )
12 mapdn0.o 
 |-  .0. = ( 0g ` U )
13 mapdn0.z 
 |-  Z = ( 0g ` C )
14 mapdn0.x 
 |-  ( ph -> X e. ( V \ { .0. } ) )
15 eldifsni 
 |-  ( X e. ( V \ { .0. } ) -> X =/= .0. )
16 14 15 syl 
 |-  ( ph -> X =/= .0. )
17 sneq 
 |-  ( F = Z -> { F } = { Z } )
18 17 fveq2d 
 |-  ( F = Z -> ( J ` { F } ) = ( J ` { Z } ) )
19 11 18 sylan9eq 
 |-  ( ( ph /\ F = Z ) -> ( M ` ( N ` { X } ) ) = ( J ` { Z } ) )
20 1 2 3 12 6 13 9 mapd0 
 |-  ( ph -> ( M ` { .0. } ) = { Z } )
21 1 6 9 lcdlmod 
 |-  ( ph -> C e. LMod )
22 13 8 lspsn0 
 |-  ( C e. LMod -> ( J ` { Z } ) = { Z } )
23 21 22 syl 
 |-  ( ph -> ( J ` { Z } ) = { Z } )
24 20 23 eqtr4d 
 |-  ( ph -> ( M ` { .0. } ) = ( J ` { Z } ) )
25 24 adantr 
 |-  ( ( ph /\ F = Z ) -> ( M ` { .0. } ) = ( J ` { Z } ) )
26 19 25 eqtr4d 
 |-  ( ( ph /\ F = Z ) -> ( M ` ( N ` { X } ) ) = ( M ` { .0. } ) )
27 26 ex 
 |-  ( ph -> ( F = Z -> ( M ` ( N ` { X } ) ) = ( M ` { .0. } ) ) )
28 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
29 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
30 14 eldifad 
 |-  ( ph -> X e. V )
31 4 28 5 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
32 29 30 31 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
33 12 28 lsssn0 
 |-  ( U e. LMod -> { .0. } e. ( LSubSp ` U ) )
34 29 33 syl 
 |-  ( ph -> { .0. } e. ( LSubSp ` U ) )
35 1 3 28 2 9 32 34 mapd11 
 |-  ( ph -> ( ( M ` ( N ` { X } ) ) = ( M ` { .0. } ) <-> ( N ` { X } ) = { .0. } ) )
36 4 12 5 lspsneq0 
 |-  ( ( U e. LMod /\ X e. V ) -> ( ( N ` { X } ) = { .0. } <-> X = .0. ) )
37 29 30 36 syl2anc 
 |-  ( ph -> ( ( N ` { X } ) = { .0. } <-> X = .0. ) )
38 35 37 bitrd 
 |-  ( ph -> ( ( M ` ( N ` { X } ) ) = ( M ` { .0. } ) <-> X = .0. ) )
39 27 38 sylibd 
 |-  ( ph -> ( F = Z -> X = .0. ) )
40 39 necon3d 
 |-  ( ph -> ( X =/= .0. -> F =/= Z ) )
41 16 40 mpd 
 |-  ( ph -> F =/= Z )
42 eldifsn 
 |-  ( F e. ( D \ { Z } ) <-> ( F e. D /\ F =/= Z ) )
43 10 41 42 sylanbrc 
 |-  ( ph -> F e. ( D \ { Z } ) )