Metamath Proof Explorer

Theorem hdmaprnlem4N

Description: Part of proof of part 12 in Baer p. 49 line 19. (T* =) (Ft)* = Gs. (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 )
hdmaprnlem1.t2 
|- ( ph -> t e. ( ( N ` { v } ) \ { .0. } ) )
Assertion hdmaprnlem4N 
|- ( ph -> ( M ` ( N ` { t } ) ) = ( L ` { 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 hdmaprnlem1.t2 
 |-  ( ph -> t e. ( ( N ` { v } ) \ { .0. } ) )
20 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
21 1 2 9 dvhlmod 
 |-  ( ph -> U e. LMod )
22 3 20 4 lspsncl 
 |-  ( ( U e. LMod /\ v e. V ) -> ( N ` { v } ) e. ( LSubSp ` U ) )
23 21 11 22 syl2anc 
 |-  ( ph -> ( N ` { v } ) e. ( LSubSp ` U ) )
24 19 eldifad 
 |-  ( ph -> t e. ( N ` { v } ) )
25 20 4 21 23 24 lspsnel5a 
 |-  ( ph -> ( N ` { t } ) C_ ( N ` { v } ) )
26 1 2 9 dvhlvec 
 |-  ( ph -> U e. LVec )
27 3 20 lss1 
 |-  ( U e. LMod -> V e. ( LSubSp ` U ) )
28 21 27 syl 
 |-  ( ph -> V e. ( LSubSp ` U ) )
29 20 4 21 28 11 lspsnel5a 
 |-  ( ph -> ( N ` { v } ) C_ V )
30 29 ssdifd 
 |-  ( ph -> ( ( N ` { v } ) \ { .0. } ) C_ ( V \ { .0. } ) )
31 30 19 sseldd 
 |-  ( ph -> t e. ( V \ { .0. } ) )
32 3 17 4 26 31 11 lspsncmp 
 |-  ( ph -> ( ( N ` { t } ) C_ ( N ` { v } ) <-> ( N ` { t } ) = ( N ` { v } ) ) )
33 25 32 mpbid 
 |-  ( ph -> ( N ` { t } ) = ( N ` { v } ) )
34 33 fveq2d 
 |-  ( ph -> ( M ` ( N ` { t } ) ) = ( M ` ( N ` { v } ) ) )
35 34 12 eqtrd 
 |-  ( ph -> ( M ` ( N ` { t } ) ) = ( L ` { s } ) )