Metamath Proof Explorer

Theorem hgmaprnlem1N

Description: Lemma for hgmaprnN . (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.k2 
|- ( ph -> k e. B )
hgmaprnlem1.sk 
|- ( ph -> s = ( k .x. t ) )
Assertion hgmaprnlem1N 
|- ( ph -> z e. ran G )

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.k2 
 |-  ( ph -> k e. B )
22 hgmaprnlem1.sk 
 |-  ( ph -> s = ( k .x. t ) )
23 22 fveq2d 
 |-  ( ph -> ( S ` s ) = ( S ` ( k .x. t ) ) )
24 18 eldifad 
 |-  ( ph -> t e. V )
25 1 2 3 6 4 5 8 12 14 15 16 24 21 hgmapvs 
 |-  ( ph -> ( S ` ( k .x. t ) ) = ( ( G ` k ) .xb ( S ` t ) ) )
26 23 20 25 3eqtr3d 
 |-  ( ph -> ( z .xb ( S ` t ) ) = ( ( G ` k ) .xb ( S ` t ) ) )
27 1 8 16 lcdlvec 
 |-  ( ph -> C e. LVec )
28 1 2 4 5 8 10 11 15 16 21 hgmapdcl 
 |-  ( ph -> ( G ` k ) e. A )
29 1 2 3 8 9 14 16 24 hdmapcl 
 |-  ( ph -> ( S ` t ) e. D )
30 eldifsni 
 |-  ( t e. ( V \ { .0. } ) -> t =/= .0. )
31 18 30 syl 
 |-  ( ph -> t =/= .0. )
32 1 2 3 7 8 13 14 16 24 hdmapeq0 
 |-  ( ph -> ( ( S ` t ) = Q <-> t = .0. ) )
33 32 necon3bid 
 |-  ( ph -> ( ( S ` t ) =/= Q <-> t =/= .0. ) )
34 31 33 mpbird 
 |-  ( ph -> ( S ` t ) =/= Q )
35 9 12 10 11 13 27 17 28 29 34 lvecvscan2 
 |-  ( ph -> ( ( z .xb ( S ` t ) ) = ( ( G ` k ) .xb ( S ` t ) ) <-> z = ( G ` k ) ) )
36 26 35 mpbid 
 |-  ( ph -> z = ( G ` k ) )
37 1 2 4 5 15 16 hgmapfnN 
 |-  ( ph -> G Fn B )
38 fnfvelrn 
 |-  ( ( G Fn B /\ k e. B ) -> ( G ` k ) e. ran G )
39 37 21 38 syl2anc 
 |-  ( ph -> ( G ` k ) e. ran G )
40 36 39 eqeltrd 
 |-  ( ph -> z e. ran G )