Metamath Proof Explorer

Theorem mapdsn

Description: Value of the map defined by df-mapd at the span of a singleton. (Contributed by NM, 16-Feb-2015)

Ref Expression
Hypotheses mapdsn.h 
|- H = ( LHyp ` K )
mapdsn.o 
|- O = ( ( ocH ` K ) ` W )
mapdsn.m 
|- M = ( ( mapd ` K ) ` W )
mapdsn.u 
|- U = ( ( DVecH ` K ) ` W )
mapdsn.v 
|- V = ( Base ` U )
mapdsn.n 
|- N = ( LSpan ` U )
mapdsn.f 
|- F = ( LFnl ` U )
mapdsn.l 
|- L = ( LKer ` U )
mapdsn.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
mapdsn.x 
|- ( ph -> X e. V )
Assertion mapdsn 
|- ( ph -> ( M ` ( N ` { X } ) ) = { f e. F | ( O ` { X } ) C_ ( L ` f ) } )

Proof

Step Hyp Ref Expression
1 mapdsn.h 
 |-  H = ( LHyp ` K )
2 mapdsn.o 
 |-  O = ( ( ocH ` K ) ` W )
3 mapdsn.m 
 |-  M = ( ( mapd ` K ) ` W )
4 mapdsn.u 
 |-  U = ( ( DVecH ` K ) ` W )
5 mapdsn.v 
 |-  V = ( Base ` U )
6 mapdsn.n 
 |-  N = ( LSpan ` U )
7 mapdsn.f 
 |-  F = ( LFnl ` U )
8 mapdsn.l 
 |-  L = ( LKer ` U )
9 mapdsn.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 mapdsn.x 
 |-  ( ph -> X e. V )
11 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
12 1 4 9 dvhlmod 
 |-  ( ph -> U e. LMod )
13 5 11 6 lspsncl 
 |-  ( ( U e. LMod /\ X e. V ) -> ( N ` { X } ) e. ( LSubSp ` U ) )
14 12 10 13 syl2anc 
 |-  ( ph -> ( N ` { X } ) e. ( LSubSp ` U ) )
15 1 4 11 7 8 2 3 9 14 mapdval 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) } )
16 9 ad2antrr 
 |-  ( ( ( ph /\ f e. F ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) ) -> ( K e. HL /\ W e. H ) )
17 10 snssd 
 |-  ( ph -> { X } C_ V )
18 5 6 lspssv 
 |-  ( ( U e. LMod /\ { X } C_ V ) -> ( N ` { X } ) C_ V )
19 12 17 18 syl2anc 
 |-  ( ph -> ( N ` { X } ) C_ V )
20 19 ad2antrr 
 |-  ( ( ( ph /\ f e. F ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) ) -> ( N ` { X } ) C_ V )
21 simprr 
 |-  ( ( ( ph /\ f e. F ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) ) -> ( O ` ( L ` f ) ) C_ ( N ` { X } ) )
22 1 4 5 2 dochss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( N ` { X } ) C_ V /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) -> ( O ` ( N ` { X } ) ) C_ ( O ` ( O ` ( L ` f ) ) ) )
23 16 20 21 22 syl3anc 
 |-  ( ( ( ph /\ f e. F ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) ) -> ( O ` ( N ` { X } ) ) C_ ( O ` ( O ` ( L ` f ) ) ) )
24 1 4 2 5 6 9 17 dochocsp 
 |-  ( ph -> ( O ` ( N ` { X } ) ) = ( O ` { X } ) )
25 24 ad2antrr 
 |-  ( ( ( ph /\ f e. F ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) ) -> ( O ` ( N ` { X } ) ) = ( O ` { X } ) )
26 simprl 
 |-  ( ( ( ph /\ f e. F ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) ) -> ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) )
27 23 25 26 3sstr3d 
 |-  ( ( ( ph /\ f e. F ) /\ ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) ) -> ( O ` { X } ) C_ ( L ` f ) )
28 9 ad2antrr 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( K e. HL /\ W e. H ) )
29 simplr 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> f e. F )
30 10 ad2antrr 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> X e. V )
31 simpr 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( O ` { X } ) C_ ( L ` f ) )
32 1 2 4 5 7 8 28 29 30 31 lcfl9a 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) )
33 12 ad2antrr 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> U e. LMod )
34 5 7 8 33 29 lkrssv 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( L ` f ) C_ V )
35 1 4 5 2 dochss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( L ` f ) C_ V /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( O ` ( L ` f ) ) C_ ( O ` ( O ` { X } ) ) )
36 28 34 31 35 syl3anc 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( O ` ( L ` f ) ) C_ ( O ` ( O ` { X } ) ) )
37 1 4 2 5 6 9 10 dochocsn 
 |-  ( ph -> ( O ` ( O ` { X } ) ) = ( N ` { X } ) )
38 37 ad2antrr 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( O ` ( O ` { X } ) ) = ( N ` { X } ) )
39 36 38 sseqtrd 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( O ` ( L ` f ) ) C_ ( N ` { X } ) )
40 32 39 jca 
 |-  ( ( ( ph /\ f e. F ) /\ ( O ` { X } ) C_ ( L ` f ) ) -> ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) )
41 27 40 impbida 
 |-  ( ( ph /\ f e. F ) -> ( ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) <-> ( O ` { X } ) C_ ( L ` f ) ) )
42 41 rabbidva 
 |-  ( ph -> { f e. F | ( ( O ` ( O ` ( L ` f ) ) ) = ( L ` f ) /\ ( O ` ( L ` f ) ) C_ ( N ` { X } ) ) } = { f e. F | ( O ` { X } ) C_ ( L ` f ) } )
43 15 42 eqtrd 
 |-  ( ph -> ( M ` ( N ` { X } ) ) = { f e. F | ( O ` { X } ) C_ ( L ` f ) } )