Metamath Proof Explorer

Theorem hdmaprnlem3N

Description: Part of proof of part 12 in Baer p. 49 line 15, T =/= P. Our (`' M `( L{ ( ( Su ) .+b s ) } ) ) is Baer's P, where P* = G(u'+s). (Contributed by NM, 27-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 hdmaprnlem3N 
|- ( ph -> ( N ` { v } ) =/= ( `' 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 1 5 9 lcdlmod 
 |-  ( ph -> C e. LMod )
20 1 2 3 5 15 8 9 13 hdmapcl 
 |-  ( ph -> ( S ` u ) e. D )
21 10 eldifad 
 |-  ( ph -> s e. D )
22 15 18 lmodvacl 
 |-  ( ( C e. LMod /\ ( S ` u ) e. D /\ s e. D ) -> ( ( S ` u ) .+b s ) e. D )
23 19 20 21 22 syl3anc 
 |-  ( ph -> ( ( S ` u ) .+b s ) e. D )
24 eqid 
 |-  ( LSubSp ` C ) = ( LSubSp ` C )
25 15 24 6 lspsncl 
 |-  ( ( C e. LMod /\ s e. D ) -> ( L ` { s } ) e. ( LSubSp ` C ) )
26 19 21 25 syl2anc 
 |-  ( ph -> ( L ` { s } ) e. ( LSubSp ` C ) )
27 15 6 lspsnid 
 |-  ( ( C e. LMod /\ s e. D ) -> s e. ( L ` { s } ) )
28 19 21 27 syl2anc 
 |-  ( ph -> s e. ( L ` { s } ) )
29 1 5 9 lcdlvec 
 |-  ( ph -> C e. LVec )
30 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
31 1 2 9 dvhlmod 
 |-  ( ph -> U e. LMod )
32 3 30 4 lspsncl 
 |-  ( ( U e. LMod /\ v e. V ) -> ( N ` { v } ) e. ( LSubSp ` U ) )
33 31 11 32 syl2anc 
 |-  ( ph -> ( N ` { v } ) e. ( LSubSp ` U ) )
34 17 30 31 33 13 14 lssneln0 
 |-  ( ph -> u e. ( V \ { .0. } ) )
35 1 2 3 17 5 16 15 8 9 34 hdmapnzcl 
 |-  ( ph -> ( S ` u ) e. ( D \ { Q } ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 hdmaprnlem1N 
 |-  ( ph -> ( L ` { ( S ` u ) } ) =/= ( L ` { s } ) )
37 15 16 6 29 35 21 36 lspsnne1 
 |-  ( ph -> -. ( S ` u ) e. ( L ` { s } ) )
38 15 18 24 19 26 28 20 37 lssvancl2 
 |-  ( ph -> -. ( ( S ` u ) .+b s ) e. ( L ` { s } ) )
39 15 6 19 23 21 38 lspsnne2 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) =/= ( L ` { s } ) )
40 39 necomd 
 |-  ( ph -> ( L ` { s } ) =/= ( L ` { ( ( S ` u ) .+b s ) } ) )
41 15 24 6 lspsncl 
 |-  ( ( C e. LMod /\ ( ( S ` u ) .+b s ) e. D ) -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
42 19 23 41 syl2anc 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ( LSubSp ` C ) )
43 1 7 5 24 9 mapdrn2 
 |-  ( ph -> ran M = ( LSubSp ` C ) )
44 42 43 eleqtrrd 
 |-  ( ph -> ( L ` { ( ( S ` u ) .+b s ) } ) e. ran M )
45 1 7 9 44 mapdcnvid2 
 |-  ( ph -> ( M ` ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) ) = ( L ` { ( ( S ` u ) .+b s ) } ) )
46 40 12 45 3netr4d 
 |-  ( ph -> ( M ` ( N ` { v } ) ) =/= ( M ` ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) ) )
47 1 7 2 30 9 44 mapdcnvcl 
 |-  ( ph -> ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) e. ( LSubSp ` U ) )
48 1 2 30 7 9 33 47 mapd11 
 |-  ( ph -> ( ( M ` ( N ` { v } ) ) = ( M ` ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) ) <-> ( N ` { v } ) = ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) ) )
49 48 necon3bid 
 |-  ( ph -> ( ( M ` ( N ` { v } ) ) =/= ( M ` ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) ) <-> ( N ` { v } ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) ) )
50 46 49 mpbid 
 |-  ( ph -> ( N ` { v } ) =/= ( `' M ` ( L ` { ( ( S ` u ) .+b s ) } ) ) )