Metamath Proof Explorer

Theorem mapdindp

Description: Transfer (part of) vector independence condition from domain to range of projectivity mapd . (Contributed by NM, 11-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 } ) )
mapdindp.x 
|- ( ph -> X e. V )
mapdindp.y 
|- ( ph -> Y e. V )
mapdindp.g 
|- ( ph -> G e. D )
mapdindp.my 
|- ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
mapdindp.z 
|- ( ph -> Z e. V )
mapdindp.e 
|- ( ph -> E e. D )
mapdindp.mg 
|- ( ph -> ( M ` ( N ` { Z } ) ) = ( J ` { E } ) )
mapdindp.xn 
|- ( ph -> -. X e. ( N ` { Y , Z } ) )
Assertion mapdindp 
|- ( ph -> -. F e. ( J ` { G , E } ) )

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 mapdindp.x 
 |-  ( ph -> X e. V )
13 mapdindp.y 
 |-  ( ph -> Y e. V )
14 mapdindp.g 
 |-  ( ph -> G e. D )
15 mapdindp.my 
 |-  ( ph -> ( M ` ( N ` { Y } ) ) = ( J ` { G } ) )
16 mapdindp.z 
 |-  ( ph -> Z e. V )
17 mapdindp.e 
 |-  ( ph -> E e. D )
18 mapdindp.mg 
 |-  ( ph -> ( M ` ( N ` { Z } ) ) = ( J ` { E } ) )
19 mapdindp.xn 
 |-  ( ph -> -. X e. ( N ` { Y , Z } ) )
20 eqid 
 |-  ( LSubSp ` C ) = ( LSubSp ` C )
21 1 6 9 lcdlmod 
 |-  ( ph -> C e. LMod )
22 7 20 8 21 14 17 lspprcl 
 |-  ( ph -> ( J ` { G , E } ) e. ( LSubSp ` C ) )
23 7 20 8 21 22 10 lspsnel5 
 |-  ( ph -> ( F e. ( J ` { G , E } ) <-> ( J ` { F } ) C_ ( J ` { G , E } ) ) )
24 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
25 1 3 9 dvhlmod 
 |-  ( ph -> U e. LMod )
26 4 24 5 25 13 16 lspprcl 
 |-  ( ph -> ( N ` { Y , Z } ) e. ( LSubSp ` U ) )
27 4 24 5 25 26 12 lspsnel5 
 |-  ( ph -> ( X e. ( N ` { Y , Z } ) <-> ( N ` { X } ) C_ ( N ` { Y , Z } ) ) )
28 4 24 5 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
29 25 12 28 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
30 1 3 24 2 9 29 26 mapdord 
 |-  ( ph -> ( ( M ` ( N ` { X } ) ) C_ ( M ` ( N ` { Y , Z } ) ) <-> ( N ` { X } ) C_ ( N ` { Y , Z } ) ) )
31 eqid 
 |-  ( LSSum ` U ) = ( LSSum ` U )
32 4 5 31 25 13 16 lsmpr 
 |-  ( ph -> ( N ` { Y , Z } ) = ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) )
33 32 fveq2d 
 |-  ( ph -> ( M ` ( N ` { Y , Z } ) ) = ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) )
34 eqid 
 |-  ( LSSum ` C ) = ( LSSum ` C )
35 4 24 5 lspsncl 
 |-  ( ( U e. LMod /\ Y e. V ) -> ( N ` { Y } ) e. ( LSubSp ` U ) )
36 25 13 35 syl2anc 
 |-  ( ph -> ( N ` { Y } ) e. ( LSubSp ` U ) )
37 4 24 5 lspsncl 
 |-  ( ( U e. LMod /\ Z e. V ) -> ( N ` { Z } ) e. ( LSubSp ` U ) )
38 25 16 37 syl2anc 
 |-  ( ph -> ( N ` { Z } ) e. ( LSubSp ` U ) )
39 1 2 3 24 31 6 34 9 36 38 mapdlsm 
 |-  ( ph -> ( M ` ( ( N ` { Y } ) ( LSSum ` U ) ( N ` { Z } ) ) ) = ( ( M ` ( N ` { Y } ) ) ( LSSum ` C ) ( M ` ( N ` { Z } ) ) ) )
40 15 18 oveq12d 
 |-  ( ph -> ( ( M ` ( N ` { Y } ) ) ( LSSum ` C ) ( M ` ( N ` { Z } ) ) ) = ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) )
41 33 39 40 3eqtrd 
 |-  ( ph -> ( M ` ( N ` { Y , Z } ) ) = ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) )
42 7 8 34 21 14 17 lsmpr 
 |-  ( ph -> ( J ` { G , E } ) = ( ( J ` { G } ) ( LSSum ` C ) ( J ` { E } ) ) )
43 41 42 eqtr4d 
 |-  ( ph -> ( M ` ( N ` { Y , Z } ) ) = ( J ` { G , E } ) )
44 11 43 sseq12d 
 |-  ( ph -> ( ( M ` ( N ` { X } ) ) C_ ( M ` ( N ` { Y , Z } ) ) <-> ( J ` { F } ) C_ ( J ` { G , E } ) ) )
45 27 30 44 3bitr2rd 
 |-  ( ph -> ( ( J ` { F } ) C_ ( J ` { G , E } ) <-> X e. ( N ` { Y , Z } ) ) )
46 23 45 bitrd 
 |-  ( ph -> ( F e. ( J ` { G , E } ) <-> X e. ( N ` { Y , Z } ) ) )
47 19 46 mtbird 
 |-  ( ph -> -. F e. ( J ` { G , E } ) )