Metamath Proof Explorer

Theorem hgmaprnlem3N

Description: Lemma for hgmaprnN . Eliminate k . (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 hgmaprnlem3N 
|- ( 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.m 
 |-  M = ( ( mapd ` K ) ` W )
22 hgmaprnlem1.n 
 |-  N = ( LSpan ` U )
23 hgmaprnlem1.l 
 |-  L = ( LSpan ` C )
24 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 hgmaprnlem2N 
 |-  ( ph -> ( N ` { s } ) C_ ( N ` { t } ) )
25 1 2 16 dvhlmod 
 |-  ( ph -> U e. LMod )
26 18 eldifad 
 |-  ( ph -> t e. V )
27 3 4 5 6 22 25 19 26 lspsnss2 
 |-  ( ph -> ( ( N ` { s } ) C_ ( N ` { t } ) <-> E. k e. B s = ( k .x. t ) ) )
28 24 27 mpbid 
 |-  ( ph -> E. k e. B s = ( k .x. t ) )
29 16 3ad2ant1 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> ( K e. HL /\ W e. H ) )
30 17 3ad2ant1 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> z e. A )
31 18 3ad2ant1 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> t e. ( V \ { .0. } ) )
32 19 3ad2ant1 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> s e. V )
33 20 3ad2ant1 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> ( S ` s ) = ( z .xb ( S ` t ) ) )
34 simp2 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> k e. B )
35 simp3 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> s = ( k .x. t ) )
36 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 29 30 31 32 33 34 35 hgmaprnlem1N 
 |-  ( ( ph /\ k e. B /\ s = ( k .x. t ) ) -> z e. ran G )
37 36 rexlimdv3a 
 |-  ( ph -> ( E. k e. B s = ( k .x. t ) -> z e. ran G ) )
38 28 37 mpd 
 |-  ( ph -> z e. ran G )