Metamath Proof Explorer

Theorem hdmaprnlem3uN

Description: Part of proof of part 12 in Baer p. 49. (Contributed by NM, 29-May-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hdmaprnlem1.h 
|- H = ( LHyp ` K )
hdmaprnlem1.u 
|- U = ( ( DVecH ` K ) ` W )
hdmaprnlem1.v 
|- V = ( Base ` U )
hdmaprnlem1.n 
|- N = ( LSpan ` U )
hdmaprnlem1.c 
|- C = ( ( LCDual ` K ) ` W )
hdmaprnlem1.l 
|- L = ( LSpan ` C )
hdmaprnlem1.m 
|- M = ( ( mapd ` K ) ` W )
hdmaprnlem1.s 
|- S = ( ( HDMap ` K ) ` W )
hdmaprnlem1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hdmaprnlem1.se 
|- ( ph -> s e. ( D \ { Q } ) )
hdmaprnlem1.ve 
|- ( ph -> v e. V )
hdmaprnlem1.e 
|- ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
hdmaprnlem1.ue 
|- ( ph -> u e. V )
hdmaprnlem1.un 
|- ( ph -> -. u e. ( N ` { v } ) )
hdmaprnlem1.d 
|- D = ( Base ` C )
hdmaprnlem1.q 
|- Q = ( 0g ` C )
hdmaprnlem1.o 
|- .0. = ( 0g ` U )
hdmaprnlem1.a 
|- .+b = ( +g ` C )
Assertion hdmaprnlem3uN 
|- ( ph -> ( N ` { u } ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) )

Proof

Step Hyp Ref Expression
1 hdmaprnlem1.h 
 |-  H = ( LHyp ` K )
2 hdmaprnlem1.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hdmaprnlem1.v 
 |-  V = ( Base ` U )
4 hdmaprnlem1.n 
 |-  N = ( LSpan ` U )
5 hdmaprnlem1.c 
 |-  C = ( ( LCDual ` K ) ` W )
6 hdmaprnlem1.l 
 |-  L = ( LSpan ` C )
7 hdmaprnlem1.m 
 |-  M = ( ( mapd ` K ) ` W )
8 hdmaprnlem1.s 
 |-  S = ( ( HDMap ` K ) ` W )
9 hdmaprnlem1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
10 hdmaprnlem1.se 
 |-  ( ph -> s e. ( D \ { Q } ) )
11 hdmaprnlem1.ve 
 |-  ( ph -> v e. V )
12 hdmaprnlem1.e 
 |-  ( ph -> ( M ` ( N ` { v } ) ) = ( L ` { s } ) )
13 hdmaprnlem1.ue 
 |-  ( ph -> u e. V )
14 hdmaprnlem1.un 
 |-  ( ph -> -. u e. ( N ` { v } ) )
15 hdmaprnlem1.d 
 |-  D = ( Base ` C )
16 hdmaprnlem1.q 
 |-  Q = ( 0g ` C )
17 hdmaprnlem1.o 
 |-  .0. = ( 0g ` U )
18 hdmaprnlem1.a 
 |-  .+b = ( +g ` C )
19 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
20 1 2 9 dvhlmod 
 |-  ( ph -> U e. LMod )
21 3 19 4 lspsncl 
 |-  ( ( U e. LMod /\ u e. V ) -> ( N ` { u } ) e. ( LSubSp ` U ) )
22 20 13 21 syl2anc 
 |-  ( ph -> ( N ` { u } ) e. ( LSubSp ` U ) )
23 1 7 2 19 9 22 mapdcnvid1N 
 |-  ( ph -> ( `' M ` ( M ` ( N ` { u } ) ) ) = ( N ` { u } ) )
24 1 2 3 4 5 6 7 8 9 13 hdmap10 
 |-  ( ph -> ( M ` ( N ` { u } ) ) = ( L ` { ( S ` u ) } ) )
25 1 5 9 lcdlvec 
 |-  ( ph -> C e. LVec )
26 1 2 3 5 15 8 9 13 hdmapcl 
 |-  ( ph -> ( S ` u ) e. D )
27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 hdmaprnlem1N 
 |-  ( ph -> ( L ` { ( S ` u ) } ) =/= ( L ` { s } ) )
28 15 18 16 6 25 26 10 27 lspindp3 
 |-  ( ph -> ( L ` { ( S ` u ) } ) =/= ( L ` { ( ( S ` u ) .+b s ) } ) )
29 24 28 eqnetrd 
 |-  ( ph -> ( M ` ( N ` { u } ) ) =/= ( L ` { ( ( S ` u ) .+b s ) } ) )
30 1 7 2 19 9 22 mapdcl 
 |-  ( ph -> ( M ` ( N ` { u } ) ) e. ran M )
31 1 5 9 lcdlmod 
 |-  ( ph -> C e. LMod )
32 10 eldifad 
 |-  ( ph -> s e. D )
33 15 18 lmodvacl 
 |-  ( ( C e. LMod /\ ( S ` u ) e. D /\ s e. D ) -> ( ( S ` u ) .+b s ) e. D )
34 31 26 32 33 syl3anc 
 |-  ( ph -> ( ( S ` u ) .+b s ) e. D )
35 eqid 
 |-  ( LSubSp ` C ) = ( LSubSp ` C )
36 15 35 6 lspsncl 
 |-  ( ( C e. LMod /\ ( ( S ` u ) .+b s ) e. D ) -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
37 31 34 36 syl2anc 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
38 1 7 5 35 9 mapdrn2 
 |-  ( ph -> ran M = ( LSubSp ` C ) )
39 37 38 eleqtrrd 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ran M )
40 1 7 9 30 39 mapdcnv11N 
 |-  ( ph -> ( ( `' M ` ( M ` ( N ` { u } ) ) ) = ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) <-> ( M ` ( N ` { u } ) ) = ( L ` { ( ( S ` u ) .+b s ) } ) ) )
41 40 necon3bid 
 |-  ( ph -> ( ( `' M ` ( M ` ( N ` { u } ) ) ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) <-> ( M ` ( N ` { u } ) ) =/= ( L ` { ( ( S ` u ) .+b s ) } ) ) )
42 29 41 mpbird 
 |-  ( ph -> ( `' M ` ( M ` ( N ` { u } ) ) ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) )
43 23 42 eqnetrrd 
 |-  ( ph -> ( N ` { u } ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) )