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Theorem hdmapval3lemN 32652
Description: Value of map from vectors to functionals at arguments not colinear with the reference vector 
E. (Contributed by NM, 17-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
hdmapval3.h  |-  H  =  ( LHyp `  K
)
hdmapval3.e  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
hdmapval3.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmapval3.v  |-  V  =  ( Base `  U
)
hdmapval3.n  |-  N  =  ( LSpan `  U )
hdmapval3.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmapval3.d  |-  D  =  ( Base `  C
)
hdmapval3.j  |-  J  =  ( (HVMap `  K
) `  W )
hdmapval3.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmapval3.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmapval3.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmapval3.te  |-  ( ph  ->  ( N `  { T } )  =/=  ( N `  { E } ) )
hdmapval3lem.t  |-  ( ph  ->  T  e.  ( V 
\  { ( 0g
`  U ) } ) )
hdmapval3lem.x  |-  ( ph  ->  x  e.  V )
hdmapval3lem.xn  |-  ( ph  ->  -.  x  e.  ( N `  { E ,  T } ) )
Assertion
Ref Expression
hdmapval3lemN  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. E ,  ( J `  E ) ,  T >. )
)

Proof of Theorem hdmapval3lemN
StepHypRef Expression
1 hdmapval3.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmapval3.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmapval3.v . . 3  |-  V  =  ( Base `  U
)
4 eqid 2296 . . 3  |-  ( 0g
`  U )  =  ( 0g `  U
)
5 hdmapval3.n . . 3  |-  N  =  ( LSpan `  U )
6 hdmapval3.c . . 3  |-  C  =  ( (LCDual `  K
) `  W )
7 hdmapval3.d . . 3  |-  D  =  ( Base `  C
)
8 eqid 2296 . . 3  |-  ( LSpan `  C )  =  (
LSpan `  C )
9 eqid 2296 . . 3  |-  ( (mapd `  K ) `  W
)  =  ( (mapd `  K ) `  W
)
10 hdmapval3.i . . 3  |-  I  =  ( (HDMap1 `  K
) `  W )
11 hdmapval3.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
12 eqid 2296 . . . . . 6  |-  ( 0g
`  C )  =  ( 0g `  C
)
13 hdmapval3.j . . . . . 6  |-  J  =  ( (HVMap `  K
) `  W )
14 eqid 2296 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
15 eqid 2296 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
16 hdmapval3.e . . . . . . 7  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
171, 14, 15, 2, 3, 4, 16, 11dvheveccl 31924 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
181, 2, 3, 4, 6, 7, 12, 13, 11, 17hvmapcl2 32578 . . . . 5  |-  ( ph  ->  ( J `  E
)  e.  ( D 
\  { ( 0g
`  C ) } ) )
19 eldifi 3311 . . . . 5  |-  ( ( J `  E )  e.  ( D  \  { ( 0g `  C ) } )  ->  ( J `  E )  e.  D
)
2018, 19syl 15 . . . 4  |-  ( ph  ->  ( J `  E
)  e.  D )
211, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17mapdhvmap 32581 . . . 4  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( N `  { E } ) )  =  ( (
LSpan `  C ) `  { ( J `  E ) } ) )
221, 2, 11dvhlvec 31921 . . . . . . 7  |-  ( ph  ->  U  e.  LVec )
23 hdmapval3lem.x . . . . . . 7  |-  ( ph  ->  x  e.  V )
24 eldifi 3311 . . . . . . . 8  |-  ( E  e.  ( V  \  { ( 0g `  U ) } )  ->  E  e.  V
)
2517, 24syl 15 . . . . . . 7  |-  ( ph  ->  E  e.  V )
26 hdmapval3lem.t . . . . . . . 8  |-  ( ph  ->  T  e.  ( V 
\  { ( 0g
`  U ) } ) )
27 eldifi 3311 . . . . . . . 8  |-  ( T  e.  ( V  \  { ( 0g `  U ) } )  ->  T  e.  V
)
2826, 27syl 15 . . . . . . 7  |-  ( ph  ->  T  e.  V )
29 hdmapval3lem.xn . . . . . . 7  |-  ( ph  ->  -.  x  e.  ( N `  { E ,  T } ) )
303, 5, 22, 23, 25, 28, 29lspindpi 15901 . . . . . 6  |-  ( ph  ->  ( ( N `  { x } )  =/=  ( N `  { E } )  /\  ( N `  { x } )  =/=  ( N `  { T } ) ) )
3130simpld 445 . . . . 5  |-  ( ph  ->  ( N `  {
x } )  =/=  ( N `  { E } ) )
3231necomd 2542 . . . 4  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { x } ) )
331, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 32, 17, 23hdmap1cl 32617 . . 3  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  e.  D )
34 eqidd 2297 . . . . 5  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  =  ( I `
 <. E ,  ( J `  E ) ,  x >. )
)
35 eqid 2296 . . . . . 6  |-  ( -g `  U )  =  (
-g `  U )
36 eqid 2296 . . . . . 6  |-  ( -g `  C )  =  (
-g `  C )
37 eqid 2296 . . . . . . 7  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
381, 2, 11dvhlmod 31922 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
393, 37, 5, 38, 25, 28lspprcl 15751 . . . . . . 7  |-  ( ph  ->  ( N `  { E ,  T }
)  e.  ( LSubSp `  U ) )
403, 4, 37, 38, 39, 23, 29lssneln0 15725 . . . . . 6  |-  ( ph  ->  x  e.  ( V 
\  { ( 0g
`  U ) } ) )
411, 2, 3, 35, 4, 5, 6, 7, 36, 8, 9, 10, 11, 17, 20, 40, 33, 32, 21hdmap1eq 32614 . . . . 5  |-  ( ph  ->  ( ( I `  <. E ,  ( J `
 E ) ,  x >. )  =  ( I `  <. E , 
( J `  E
) ,  x >. )  <-> 
( ( ( (mapd `  K ) `  W
) `  ( N `  { x } ) )  =  ( (
LSpan `  C ) `  { ( I `  <. E ,  ( J `
 E ) ,  x >. ) } )  /\  ( ( (mapd `  K ) `  W
) `  ( N `  { ( E (
-g `  U )
x ) } ) )  =  ( (
LSpan `  C ) `  { ( ( J `
 E ) (
-g `  C )
( I `  <. E ,  ( J `  E ) ,  x >. ) ) } ) ) ) )
4234, 41mpbid 201 . . . 4  |-  ( ph  ->  ( ( ( (mapd `  K ) `  W
) `  ( N `  { x } ) )  =  ( (
LSpan `  C ) `  { ( I `  <. E ,  ( J `
 E ) ,  x >. ) } )  /\  ( ( (mapd `  K ) `  W
) `  ( N `  { ( E (
-g `  U )
x ) } ) )  =  ( (
LSpan `  C ) `  { ( ( J `
 E ) (
-g `  C )
( I `  <. E ,  ( J `  E ) ,  x >. ) ) } ) ) )
4342simpld 445 . . 3  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( N `  { x } ) )  =  ( (
LSpan `  C ) `  { ( I `  <. E ,  ( J `
 E ) ,  x >. ) } ) )
44 hdmapval3.te . . . 4  |-  ( ph  ->  ( N `  { T } )  =/=  ( N `  { E } ) )
4544necomd 2542 . . 3  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { T } ) )
46 hdmapval3.s . . . . 5  |-  S  =  ( (HDMap `  K
) `  W )
473, 5, 38, 25, 28lspprid1 15770 . . . . . . . . 9  |-  ( ph  ->  E  e.  ( N `
 { E ,  T } ) )
4837, 5, 38, 39, 47lspsnel5a 15769 . . . . . . . 8  |-  ( ph  ->  ( N `  { E } )  C_  ( N `  { E ,  T } ) )
4948, 48unssd 3364 . . . . . . 7  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { E } ) )  C_  ( N `  { E ,  T } ) )
5049sseld 3192 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( N `  { E } )  u.  ( N `  { E } ) )  ->  x  e.  ( N `  { E ,  T } ) ) )
5129, 50mtod 168 . . . . 5  |-  ( ph  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { E } ) ) )
521, 16, 2, 3, 5, 6, 7, 13, 10, 46, 11, 25, 23, 51hdmapval2 32647 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  E >. )
)
531, 16, 13, 46, 11hdmapevec 32650 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( J `
 E ) )
5452, 53eqtr3d 2330 . . 3  |-  ( ph  ->  ( I `  <. x ,  ( I `  <. E ,  ( J `
 E ) ,  x >. ) ,  E >. )  =  ( J `
 E ) )
553, 5, 38, 25, 28lspprid2 15771 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( N `
 { E ,  T } ) )
5637, 5, 38, 39, 55lspsnel5a 15769 . . . . . . . 8  |-  ( ph  ->  ( N `  { T } )  C_  ( N `  { E ,  T } ) )
5748, 56unssd 3364 . . . . . . 7  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { T } ) )  C_  ( N `  { E ,  T } ) )
5857sseld 3192 . . . . . 6  |-  ( ph  ->  ( x  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  ->  x  e.  ( N `  { E ,  T } ) ) )
5929, 58mtod 168 . . . . 5  |-  ( ph  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) )
601, 16, 2, 3, 5, 6, 7, 13, 10, 46, 11, 28, 23, 59hdmapval2 32647 . . . 4  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  T >. )
)
6160eqcomd 2301 . . 3  |-  ( ph  ->  ( I `  <. x ,  ( I `  <. E ,  ( J `
 E ) ,  x >. ) ,  T >. )  =  ( S `
 T ) )
621, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 33, 43, 40, 17, 26, 45, 29, 54, 61hdmap1eq4N 32619 . 2  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  T >. )  =  ( S `
 T ) )
6362eqcomd 2301 1  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. E ,  ( J `  E ) ,  T >. )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162    u. cun 3163   {csn 3653   {cpr 3654   <.cop 3656   <.cotp 3657    _I cid 4320    |` cres 4707   ` cfv 5271  (class class class)co 5874   Basecbs 13164   0gc0g 13416   -gcsg 14381   LSubSpclss 15705   LSpanclspn 15744   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   DVecHcdvh 31890  LCDualclcd 32398  mapdcmpd 32436  HVMapchvm 32568  HDMap1chdma1 32604  HDMapchdma 32605
This theorem is referenced by:  hdmapval3N  32653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-oppg 14835  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lshyp 29789  df-lcv 29831  df-lfl 29870  df-lkr 29898  df-ldual 29936  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tgrp 31554  df-tendo 31566  df-edring 31568  df-dveca 31814  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041  df-doch 32160  df-djh 32207  df-lcdual 32399  df-mapd 32437  df-hvmap 32569  df-hdmap1 32606  df-hdmap 32607
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