Metamath Proof Explorer

Theorem hgmaprnlem2N

Description: Lemma for hgmaprnN . Part 15 of Baer p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero z is taken care of automatically. (Contributed by NM, 7-Jun-2015) (New usage is discouraged.)

Ref Expression
Hypotheses hgmaprnlem1.h 
|- H = ( LHyp ` K )
hgmaprnlem1.u 
|- U = ( ( DVecH ` K ) ` W )
hgmaprnlem1.v 
|- V = ( Base ` U )
hgmaprnlem1.r 
|- R = ( Scalar ` U )
hgmaprnlem1.b 
|- B = ( Base ` R )
hgmaprnlem1.t 
|- .x. = ( .s ` U )
hgmaprnlem1.o 
|- .0. = ( 0g ` U )
hgmaprnlem1.c 
|- C = ( ( LCDual ` K ) ` W )
hgmaprnlem1.d 
|- D = ( Base ` C )
hgmaprnlem1.p 
|- P = ( Scalar ` C )
hgmaprnlem1.a 
|- A = ( Base ` P )
hgmaprnlem1.e 
|- .xb = ( .s ` C )
hgmaprnlem1.q 
|- Q = ( 0g ` C )
hgmaprnlem1.s 
|- S = ( ( HDMap ` K ) ` W )
hgmaprnlem1.g 
|- G = ( ( HGMap ` K ) ` W )
hgmaprnlem1.k 
|- ( ph -> ( K e. HL /\ W e. H ) )
hgmaprnlem1.z 
|- ( ph -> z e. A )
hgmaprnlem1.t2 
|- ( ph -> t e. ( V \ { .0. } ) )
hgmaprnlem1.s2 
|- ( ph -> s e. V )
hgmaprnlem1.sz 
|- ( ph -> ( S ` s ) = ( z .xb ( S ` t ) ) )
hgmaprnlem1.m 
|- M = ( ( mapd ` K ) ` W )
hgmaprnlem1.n 
|- N = ( LSpan ` U )
hgmaprnlem1.l 
|- L = ( LSpan ` C )
Assertion hgmaprnlem2N 
|- ( ph -> ( N ` { s } ) C_ ( N ` { t } ) )

Proof

Step Hyp Ref Expression
1 hgmaprnlem1.h 
 |-  H = ( LHyp ` K )
2 hgmaprnlem1.u 
 |-  U = ( ( DVecH ` K ) ` W )
3 hgmaprnlem1.v 
 |-  V = ( Base ` U )
4 hgmaprnlem1.r 
 |-  R = ( Scalar ` U )
5 hgmaprnlem1.b 
 |-  B = ( Base ` R )
6 hgmaprnlem1.t 
 |-  .x. = ( .s ` U )
7 hgmaprnlem1.o 
 |-  .0. = ( 0g ` U )
8 hgmaprnlem1.c 
 |-  C = ( ( LCDual ` K ) ` W )
9 hgmaprnlem1.d 
 |-  D = ( Base ` C )
10 hgmaprnlem1.p 
 |-  P = ( Scalar ` C )
11 hgmaprnlem1.a 
 |-  A = ( Base ` P )
12 hgmaprnlem1.e 
 |-  .xb = ( .s ` C )
13 hgmaprnlem1.q 
 |-  Q = ( 0g ` C )
14 hgmaprnlem1.s 
 |-  S = ( ( HDMap ` K ) ` W )
15 hgmaprnlem1.g 
 |-  G = ( ( HGMap ` K ) ` W )
16 hgmaprnlem1.k 
 |-  ( ph -> ( K e. HL /\ W e. H ) )
17 hgmaprnlem1.z 
 |-  ( ph -> z e. A )
18 hgmaprnlem1.t2 
 |-  ( ph -> t e. ( V \ { .0. } ) )
19 hgmaprnlem1.s2 
 |-  ( ph -> s e. V )
20 hgmaprnlem1.sz 
 |-  ( ph -> ( S ` s ) = ( z .xb ( S ` t ) ) )
21 hgmaprnlem1.m 
 |-  M = ( ( mapd ` K ) ` W )
22 hgmaprnlem1.n 
 |-  N = ( LSpan ` U )
23 hgmaprnlem1.l 
 |-  L = ( LSpan ` C )
24 1 8 16 lcdlmod 
 |-  ( ph -> C e. LMod )
25 18 eldifad 
 |-  ( ph -> t e. V )
26 1 2 3 8 9 14 16 25 hdmapcl 
 |-  ( ph -> ( S ` t ) e. D )
27 10 11 9 12 23 lspsnvsi 
 |-  ( ( C e. LMod /\ z e. A /\ ( S ` t ) e. D ) -> ( L ` { ( z .xb ( S ` t ) ) } ) C_ ( L ` { ( S ` t ) } ) )
28 24 17 26 27 syl3anc 
 |-  ( ph -> ( L ` { ( z .xb ( S ` t ) ) } ) C_ ( L ` { ( S ` t ) } ) )
29 1 2 3 22 8 23 21 14 16 19 hdmap10 
 |-  ( ph -> ( M ` ( N ` { s } ) ) = ( L ` { ( S ` s ) } ) )
30 20 sneqd 
 |-  ( ph -> { ( S ` s ) } = { ( z .xb ( S ` t ) ) } )
31 30 fveq2d 
 |-  ( ph -> ( L ` { ( S ` s ) } ) = ( L ` { ( z .xb ( S ` t ) ) } ) )
32 29 31 eqtrd 
 |-  ( ph -> ( M ` ( N ` { s } ) ) = ( L ` { ( z .xb ( S ` t ) ) } ) )
33 1 2 3 22 8 23 21 14 16 25 hdmap10 
 |-  ( ph -> ( M ` ( N ` { t } ) ) = ( L ` { ( S ` t ) } ) )
34 28 32 33 3sstr4d 
 |-  ( ph -> ( M ` ( N ` { s } ) ) C_ ( M ` ( N ` { t } ) ) )
35 eqid 
 |-  ( LSubSp ` U ) = ( LSubSp ` U )
36 1 2 16 dvhlmod 
 |-  ( ph -> U e. LMod )
37 3 35 22 lspsncl 
 |-  ( ( U e. LMod /\ s e. V ) -> ( N ` { s } ) e. ( LSubSp ` U ) )
38 36 19 37 syl2anc 
 |-  ( ph -> ( N ` { s } ) e. ( LSubSp ` U ) )
39 3 35 22 lspsncl 
 |-  ( ( U e. LMod /\ t e. V ) -> ( N ` { t } ) e. ( LSubSp ` U ) )
40 36 25 39 syl2anc 
 |-  ( ph -> ( N ` { t } ) e. ( LSubSp ` U ) )
41 1 2 35 21 16 38 40 mapdord 
 |-  ( ph -> ( ( M ` ( N ` { s } ) ) C_ ( M ` ( N ` { t } ) ) <-> ( N ` { s } ) C_ ( N ` { t } ) ) )
42 34 41 mpbid 
 |-  ( ph -> ( N ` { s } ) C_ ( N ` { t } ) )