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Theorem hdmapval3lemN 32575
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 2435 . . 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 2435 . . 3  |-  ( LSpan `  C )  =  (
LSpan `  C )
9 eqid 2435 . . 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 2435 . . . . . 6  |-  ( 0g
`  C )  =  ( 0g `  C
)
13 hdmapval3.j . . . . . 6  |-  J  =  ( (HVMap `  K
) `  W )
14 eqid 2435 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
15 eqid 2435 . . . . . . 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 31847 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
181, 2, 3, 4, 6, 7, 12, 13, 11, 17hvmapcl2 32501 . . . . 5  |-  ( ph  ->  ( J `  E
)  e.  ( D 
\  { ( 0g
`  C ) } ) )
1918eldifad 3324 . . . 4  |-  ( ph  ->  ( J `  E
)  e.  D )
201, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17mapdhvmap 32504 . . . 4  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( N `  { E } ) )  =  ( (
LSpan `  C ) `  { ( J `  E ) } ) )
211, 2, 11dvhlvec 31844 . . . . . . 7  |-  ( ph  ->  U  e.  LVec )
22 hdmapval3lem.x . . . . . . 7  |-  ( ph  ->  x  e.  V )
2317eldifad 3324 . . . . . . 7  |-  ( ph  ->  E  e.  V )
24 hdmapval3lem.t . . . . . . . 8  |-  ( ph  ->  T  e.  ( V 
\  { ( 0g
`  U ) } ) )
2524eldifad 3324 . . . . . . 7  |-  ( ph  ->  T  e.  V )
26 hdmapval3lem.xn . . . . . . 7  |-  ( ph  ->  -.  x  e.  ( N `  { E ,  T } ) )
273, 5, 21, 22, 23, 25, 26lspindpi 16196 . . . . . 6  |-  ( ph  ->  ( ( N `  { x } )  =/=  ( N `  { E } )  /\  ( N `  { x } )  =/=  ( N `  { T } ) ) )
2827simpld 446 . . . . 5  |-  ( ph  ->  ( N `  {
x } )  =/=  ( N `  { E } ) )
2928necomd 2681 . . . 4  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { x } ) )
301, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22hdmap1cl 32540 . . 3  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  e.  D )
31 eqidd 2436 . . . . 5  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  =  ( I `
 <. E ,  ( J `  E ) ,  x >. )
)
32 eqid 2435 . . . . . 6  |-  ( -g `  U )  =  (
-g `  U )
33 eqid 2435 . . . . . 6  |-  ( -g `  C )  =  (
-g `  C )
34 eqid 2435 . . . . . . 7  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
351, 2, 11dvhlmod 31845 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
363, 34, 5, 35, 23, 25lspprcl 16046 . . . . . . 7  |-  ( ph  ->  ( N `  { E ,  T }
)  e.  ( LSubSp `  U ) )
373, 4, 34, 35, 36, 22, 26lssneln0 16020 . . . . . 6  |-  ( ph  ->  x  e.  ( V 
\  { ( 0g
`  U ) } ) )
381, 2, 3, 32, 4, 5, 6, 7, 33, 8, 9, 10, 11, 17, 19, 37, 30, 29, 20hdmap1eq 32537 . . . . 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 >. ) ) } ) ) ) )
3931, 38mpbid 202 . . . 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 >. ) ) } ) ) )
4039simpld 446 . . 3  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( N `  { x } ) )  =  ( (
LSpan `  C ) `  { ( I `  <. E ,  ( J `
 E ) ,  x >. ) } ) )
41 hdmapval3.te . . . 4  |-  ( ph  ->  ( N `  { T } )  =/=  ( N `  { E } ) )
4241necomd 2681 . . 3  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { T } ) )
43 hdmapval3.s . . . . 5  |-  S  =  ( (HDMap `  K
) `  W )
443, 5, 35, 23, 25lspprid1 16065 . . . . . . . 8  |-  ( ph  ->  E  e.  ( N `
 { E ,  T } ) )
4534, 5, 35, 36, 44lspsnel5a 16064 . . . . . . 7  |-  ( ph  ->  ( N `  { E } )  C_  ( N `  { E ,  T } ) )
4645, 45unssd 3515 . . . . . 6  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { E } ) )  C_  ( N `  { E ,  T } ) )
4746, 26ssneldd 3343 . . . . 5  |-  ( ph  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { E } ) ) )
481, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47hdmapval2 32570 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  E >. )
)
491, 16, 13, 43, 11hdmapevec 32573 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( J `
 E ) )
5048, 49eqtr3d 2469 . . 3  |-  ( ph  ->  ( I `  <. x ,  ( I `  <. E ,  ( J `
 E ) ,  x >. ) ,  E >. )  =  ( J `
 E ) )
513, 5, 35, 23, 25lspprid2 16066 . . . . . . . 8  |-  ( ph  ->  T  e.  ( N `
 { E ,  T } ) )
5234, 5, 35, 36, 51lspsnel5a 16064 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  C_  ( N `  { E ,  T } ) )
5345, 52unssd 3515 . . . . . 6  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { T } ) )  C_  ( N `  { E ,  T } ) )
5453, 26ssneldd 3343 . . . . 5  |-  ( ph  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) )
551, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54hdmapval2 32570 . . . 4  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  T >. )
)
5655eqcomd 2440 . . 3  |-  ( ph  ->  ( I `  <. x ,  ( I `  <. E ,  ( J `
 E ) ,  x >. ) ,  T >. )  =  ( S `
 T ) )
571, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 30, 40, 37, 17, 24, 42, 26, 50, 56hdmap1eq4N 32542 . 2  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  T >. )  =  ( S `
 T ) )
5857eqcomd 2440 1  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. E ,  ( J `  E ) ,  T >. )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598    \ cdif 3309    u. cun 3310   {csn 3806   {cpr 3807   <.cop 3809   <.cotp 3810    _I cid 4485    |` cres 4872   ` cfv 5446  (class class class)co 6073   Basecbs 13461   0gc0g 13715   -gcsg 14680   LSubSpclss 16000   LSpanclspn 16039   HLchlt 30085   LHypclh 30718   LTrncltrn 30835   DVecHcdvh 31813  LCDualclcd 32321  mapdcmpd 32359  HVMapchvm 32491  HDMap1chdma1 32527  HDMapchdma 32528
This theorem is referenced by:  hdmapval3N  32576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-fal 1329  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-tpos 6471  df-undef 6535  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-0g 13719  df-mre 13803  df-mrc 13804  df-acs 13806  df-poset 14395  df-plt 14407  df-lub 14423  df-glb 14424  df-join 14425  df-meet 14426  df-p0 14460  df-p1 14461  df-lat 14467  df-clat 14529  df-mnd 14682  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-subg 14933  df-cntz 15108  df-oppg 15134  df-lsm 15262  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-oppr 15720  df-dvdsr 15738  df-unit 15739  df-invr 15769  df-dvr 15780  df-drng 15829  df-lmod 15944  df-lss 16001  df-lsp 16040  df-lvec 16167  df-lsatoms 29711  df-lshyp 29712  df-lcv 29754  df-lfl 29793  df-lkr 29821  df-ldual 29859  df-oposet 29911  df-ol 29913  df-oml 29914  df-covers 30001  df-ats 30002  df-atl 30033  df-cvlat 30057  df-hlat 30086  df-llines 30232  df-lplanes 30233  df-lvols 30234  df-lines 30235  df-psubsp 30237  df-pmap 30238  df-padd 30530  df-lhyp 30722  df-laut 30723  df-ldil 30838  df-ltrn 30839  df-trl 30893  df-tgrp 31477  df-tendo 31489  df-edring 31491  df-dveca 31737  df-disoa 31764  df-dvech 31814  df-dib 31874  df-dic 31908  df-dih 31964  df-doch 32083  df-djh 32130  df-lcdual 32322  df-mapd 32360  df-hvmap 32492  df-hdmap1 32529  df-hdmap 32530
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