Metamath Proof Explorer

Theorem hgmaprnlem4N

Description: Lemma for hgmaprnN . Eliminate s . (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. } ) )
Assertion hgmaprnlem4N 
|- ( 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 1 8 16 lcdlmod 
 |-  ( ph -> C e. LMod )
20 18 eldifad 
 |-  ( ph -> t e. V )
21 1 2 3 8 9 14 16 20 hdmapcl 
 |-  ( ph -> ( S ` t ) e. D )
22 9 10 12 11 lmodvscl 
 |-  ( ( C e. LMod /\ z e. A /\ ( S ` t ) e. D ) -> ( z .xb ( S ` t ) ) e. D )
23 19 17 21 22 syl3anc 
 |-  ( ph -> ( z .xb ( S ` t ) ) e. D )
24 1 8 9 14 16 hdmaprnN 
 |-  ( ph -> ran S = D )
25 23 24 eleqtrrd 
 |-  ( ph -> ( z .xb ( S ` t ) ) e. ran S )
26 1 2 3 14 16 hdmapfnN 
 |-  ( ph -> S Fn V )
27 fvelrnb 
 |-  ( S Fn V -> ( ( z .xb ( S ` t ) ) e. ran S <-> E. s e. V ( S ` s ) = ( z .xb ( S ` t ) ) ) )
28 26 27 syl 
 |-  ( ph -> ( ( z .xb ( S ` t ) ) e. ran S <-> E. s e. V ( S ` s ) = ( z .xb ( S ` t ) ) ) )
29 25 28 mpbid 
 |-  ( ph -> E. s e. V ( S ` s ) = ( z .xb ( S ` t ) ) )
30 16 3ad2ant1 
 |-  ( ( ph /\ s e. V /\ ( S ` s ) = ( z .xb ( S ` t ) ) ) -> ( K e. HL /\ W e. H ) )
31 17 3ad2ant1 
 |-  ( ( ph /\ s e. V /\ ( S ` s ) = ( z .xb ( S ` t ) ) ) -> z e. A )
32 18 3ad2ant1 
 |-  ( ( ph /\ s e. V /\ ( S ` s ) = ( z .xb ( S ` t ) ) ) -> t e. ( V \ { .0. } ) )
33 simp2 
 |-  ( ( ph /\ s e. V /\ ( S ` s ) = ( z .xb ( S ` t ) ) ) -> s e. V )
34 simp3 
 |-  ( ( ph /\ s e. V /\ ( S ` s ) = ( z .xb ( S ` t ) ) ) -> ( S ` s ) = ( z .xb ( S ` t ) ) )
35 eqid 
 |-  ( ( mapd ` K ) ` W ) = ( ( mapd ` K ) ` W )
36 eqid 
 |-  ( LSpan ` U ) = ( LSpan ` U )
37 eqid 
 |-  ( LSpan ` C ) = ( LSpan ` C )
38 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 30 31 32 33 34 35 36 37 hgmaprnlem3N 
 |-  ( ( ph /\ s e. V /\ ( S ` s ) = ( z .xb ( S ` t ) ) ) -> z e. ran G )
39 38 rexlimdv3a 
 |-  ( ph -> ( E. s e. V ( S ` s ) = ( z .xb ( S ` t ) ) -> z e. ran G ) )
40 29 39 mpd 
 |-  ( ph -> z e. ran G )