Metamath Proof Explorer

Theorem hdmaprnlem1N

Description: Part of proof of part 12 in Baer p. 49 line 10, Gu' =/= Gs. Our ( N{ v } ) is Baer's T. (Contributed by NM, 26-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 } ) )
Assertion hdmaprnlem1N 
|- ( ph -> ( L ` { ( S ` u ) } ) =/= ( 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 1 2 9 dvhlmod 
 |-  ( ph -> U e. LMod )
16 3 4 15 13 11 14 lspsnne2 
 |-  ( ph -> ( N ` { u } ) =/= ( N ` { v } ) )
17 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
18 3 17 4 lspsncl 
 |-  ( ( U e. LMod /\ u e. V ) -> ( N ` { u } ) e. ( LSubSp ` U ) )
19 15 13 18 syl2anc 
 |-  ( ph -> ( N ` { u } ) e. ( LSubSp ` U ) )
20 3 17 4 lspsncl 
 |-  ( ( U e. LMod /\ v e. V ) -> ( N ` { v } ) e. ( LSubSp ` U ) )
21 15 11 20 syl2anc 
 |-  ( ph -> ( N ` { v } ) e. ( LSubSp ` U ) )
22 1 2 17 7 9 19 21 mapd11 
 |-  ( ph -> ( ( M ` ( N ` { u } ) ) = ( M ` ( N ` { v } ) ) <-> ( N ` { u } ) = ( N ` { v } ) ) )
23 22 necon3bid 
 |-  ( ph -> ( ( M ` ( N ` { u } ) ) =/= ( M ` ( N ` { v } ) ) <-> ( N ` { u } ) =/= ( N ` { v } ) ) )
24 16 23 mpbird 
 |-  ( ph -> ( M ` ( N ` { u } ) ) =/= ( M ` ( N ` { v } ) ) )
25 1 2 3 4 5 6 7 8 9 13 hdmap10 
 |-  ( ph -> ( M ` ( N ` { u } ) ) = ( L ` { ( S ` u ) } ) )
26 24 25 12 3netr3d 
 |-  ( ph -> ( L ` { ( S ` u ) } ) =/= ( L ` { s } ) )