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Theorem hdmapval3lemN 32006
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 2380 . . 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 2380 . . 3  |-  ( LSpan `  C )  =  (
LSpan `  C )
9 eqid 2380 . . 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 2380 . . . . . 6  |-  ( 0g
`  C )  =  ( 0g `  C
)
13 hdmapval3.j . . . . . 6  |-  J  =  ( (HVMap `  K
) `  W )
14 eqid 2380 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
15 eqid 2380 . . . . . . 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 31278 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
181, 2, 3, 4, 6, 7, 12, 13, 11, 17hvmapcl2 31932 . . . . 5  |-  ( ph  ->  ( J `  E
)  e.  ( D 
\  { ( 0g
`  C ) } ) )
1918eldifad 3268 . . . 4  |-  ( ph  ->  ( J `  E
)  e.  D )
201, 2, 3, 4, 5, 6, 8, 9, 13, 11, 17mapdhvmap 31935 . . . 4  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( N `  { E } ) )  =  ( (
LSpan `  C ) `  { ( J `  E ) } ) )
211, 2, 11dvhlvec 31275 . . . . . . 7  |-  ( ph  ->  U  e.  LVec )
22 hdmapval3lem.x . . . . . . 7  |-  ( ph  ->  x  e.  V )
2317eldifad 3268 . . . . . . 7  |-  ( ph  ->  E  e.  V )
24 hdmapval3lem.t . . . . . . . 8  |-  ( ph  ->  T  e.  ( V 
\  { ( 0g
`  U ) } ) )
2524eldifad 3268 . . . . . . 7  |-  ( ph  ->  T  e.  V )
26 hdmapval3lem.xn . . . . . . 7  |-  ( ph  ->  -.  x  e.  ( N `  { E ,  T } ) )
273, 5, 21, 22, 23, 25, 26lspindpi 16124 . . . . . 6  |-  ( ph  ->  ( ( N `  { x } )  =/=  ( N `  { E } )  /\  ( N `  { x } )  =/=  ( N `  { T } ) ) )
2827simpld 446 . . . . 5  |-  ( ph  ->  ( N `  {
x } )  =/=  ( N `  { E } ) )
2928necomd 2626 . . . 4  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { x } ) )
301, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 19, 20, 29, 17, 22hdmap1cl 31971 . . 3  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  e.  D )
31 eqidd 2381 . . . . 5  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  =  ( I `
 <. E ,  ( J `  E ) ,  x >. )
)
32 eqid 2380 . . . . . 6  |-  ( -g `  U )  =  (
-g `  U )
33 eqid 2380 . . . . . 6  |-  ( -g `  C )  =  (
-g `  C )
34 eqid 2380 . . . . . . 7  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
351, 2, 11dvhlmod 31276 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
363, 34, 5, 35, 23, 25lspprcl 15974 . . . . . . 7  |-  ( ph  ->  ( N `  { E ,  T }
)  e.  ( LSubSp `  U ) )
373, 4, 34, 35, 36, 22, 26lssneln0 15948 . . . . . 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 31968 . . . . 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 2626 . . 3  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { T } ) )
43 hdmapval3.s . . . . 5  |-  S  =  ( (HDMap `  K
) `  W )
443, 5, 35, 23, 25lspprid1 15993 . . . . . . . 8  |-  ( ph  ->  E  e.  ( N `
 { E ,  T } ) )
4534, 5, 35, 36, 44lspsnel5a 15992 . . . . . . 7  |-  ( ph  ->  ( N `  { E } )  C_  ( N `  { E ,  T } ) )
4645, 45unssd 3459 . . . . . 6  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { E } ) )  C_  ( N `  { E ,  T } ) )
4746, 26ssneldd 3287 . . . . 5  |-  ( ph  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { E } ) ) )
481, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 23, 22, 47hdmapval2 32001 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  E >. )
)
491, 16, 13, 43, 11hdmapevec 32004 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( J `
 E ) )
5048, 49eqtr3d 2414 . . 3  |-  ( ph  ->  ( I `  <. x ,  ( I `  <. E ,  ( J `
 E ) ,  x >. ) ,  E >. )  =  ( J `
 E ) )
513, 5, 35, 23, 25lspprid2 15994 . . . . . . . 8  |-  ( ph  ->  T  e.  ( N `
 { E ,  T } ) )
5234, 5, 35, 36, 51lspsnel5a 15992 . . . . . . 7  |-  ( ph  ->  ( N `  { T } )  C_  ( N `  { E ,  T } ) )
5345, 52unssd 3459 . . . . . 6  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { T } ) )  C_  ( N `  { E ,  T } ) )
5453, 26ssneldd 3287 . . . . 5  |-  ( ph  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) )
551, 16, 2, 3, 5, 6, 7, 13, 10, 43, 11, 25, 22, 54hdmapval2 32001 . . . 4  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  T >. )
)
5655eqcomd 2385 . . 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 31973 . 2  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  T >. )  =  ( S `
 T ) )
5857eqcomd 2385 1  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. E ,  ( J `  E ) ,  T >. )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543    \ cdif 3253    u. cun 3254   {csn 3750   {cpr 3751   <.cop 3753   <.cotp 3754    _I cid 4427    |` cres 4813   ` cfv 5387  (class class class)co 6013   Basecbs 13389   0gc0g 13643   -gcsg 14608   LSubSpclss 15928   LSpanclspn 15967   HLchlt 29516   LHypclh 30149   LTrncltrn 30266   DVecHcdvh 31244  LCDualclcd 31752  mapdcmpd 31790  HVMapchvm 31922  HDMap1chdma1 31958  HDMapchdma 31959
This theorem is referenced by:  hdmapval3N  32007
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-fal 1326  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-ot 3760  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-tpos 6408  df-undef 6472  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-n0 10147  df-z 10208  df-uz 10414  df-fz 10969  df-struct 13391  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-ress 13396  df-plusg 13462  df-mulr 13463  df-sca 13465  df-vsca 13466  df-0g 13647  df-mre 13731  df-mrc 13732  df-acs 13734  df-poset 14323  df-plt 14335  df-lub 14351  df-glb 14352  df-join 14353  df-meet 14354  df-p0 14388  df-p1 14389  df-lat 14395  df-clat 14457  df-mnd 14610  df-submnd 14659  df-grp 14732  df-minusg 14733  df-sbg 14734  df-subg 14861  df-cntz 15036  df-oppg 15062  df-lsm 15190  df-cmn 15334  df-abl 15335  df-mgp 15569  df-rng 15583  df-ur 15585  df-oppr 15648  df-dvdsr 15666  df-unit 15667  df-invr 15697  df-dvr 15708  df-drng 15757  df-lmod 15872  df-lss 15929  df-lsp 15968  df-lvec 16095  df-lsatoms 29142  df-lshyp 29143  df-lcv 29185  df-lfl 29224  df-lkr 29252  df-ldual 29290  df-oposet 29342  df-ol 29344  df-oml 29345  df-covers 29432  df-ats 29433  df-atl 29464  df-cvlat 29488  df-hlat 29517  df-llines 29663  df-lplanes 29664  df-lvols 29665  df-lines 29666  df-psubsp 29668  df-pmap 29669  df-padd 29961  df-lhyp 30153  df-laut 30154  df-ldil 30269  df-ltrn 30270  df-trl 30324  df-tgrp 30908  df-tendo 30920  df-edring 30922  df-dveca 31168  df-disoa 31195  df-dvech 31245  df-dib 31305  df-dic 31339  df-dih 31395  df-doch 31514  df-djh 31561  df-lcdual 31753  df-mapd 31791  df-hvmap 31923  df-hdmap1 31960  df-hdmap 31961
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