Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  hdmapval3lemN Unicode version

Theorem hdmapval3lemN 32099
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 2358 . . 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 2358 . . 3  |-  ( LSpan `  C )  =  (
LSpan `  C )
9 eqid 2358 . . 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 2358 . . . . . 6  |-  ( 0g
`  C )  =  ( 0g `  C
)
13 hdmapval3.j . . . . . 6  |-  J  =  ( (HVMap `  K
) `  W )
14 eqid 2358 . . . . . . 7  |-  ( Base `  K )  =  (
Base `  K )
15 eqid 2358 . . . . . . 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 31371 . . . . . 6  |-  ( ph  ->  E  e.  ( V 
\  { ( 0g
`  U ) } ) )
181, 2, 3, 4, 6, 7, 12, 13, 11, 17hvmapcl2 32025 . . . . 5  |-  ( ph  ->  ( J `  E
)  e.  ( D 
\  { ( 0g
`  C ) } ) )
19 eldifi 3374 . . . . 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 32028 . . . 4  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( N `  { E } ) )  =  ( (
LSpan `  C ) `  { ( J `  E ) } ) )
221, 2, 11dvhlvec 31368 . . . . . . 7  |-  ( ph  ->  U  e.  LVec )
23 hdmapval3lem.x . . . . . . 7  |-  ( ph  ->  x  e.  V )
24 eldifi 3374 . . . . . . . 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 3374 . . . . . . . 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 15984 . . . . . 6  |-  ( ph  ->  ( ( N `  { x } )  =/=  ( N `  { E } )  /\  ( N `  { x } )  =/=  ( N `  { T } ) ) )
3130simpld 445 . . . . 5  |-  ( ph  ->  ( N `  {
x } )  =/=  ( N `  { E } ) )
3231necomd 2604 . . . 4  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { x } ) )
331, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 21, 32, 17, 23hdmap1cl 32064 . . 3  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  e.  D )
34 eqidd 2359 . . . . 5  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  x >. )  =  ( I `
 <. E ,  ( J `  E ) ,  x >. )
)
35 eqid 2358 . . . . . 6  |-  ( -g `  U )  =  (
-g `  U )
36 eqid 2358 . . . . . 6  |-  ( -g `  C )  =  (
-g `  C )
37 eqid 2358 . . . . . . 7  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
381, 2, 11dvhlmod 31369 . . . . . . 7  |-  ( ph  ->  U  e.  LMod )
393, 37, 5, 38, 25, 28lspprcl 15834 . . . . . . 7  |-  ( ph  ->  ( N `  { E ,  T }
)  e.  ( LSubSp `  U ) )
403, 4, 37, 38, 39, 23, 29lssneln0 15808 . . . . . 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 32061 . . . . 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 2604 . . 3  |-  ( ph  ->  ( N `  { E } )  =/=  ( N `  { T } ) )
46 hdmapval3.s . . . . 5  |-  S  =  ( (HDMap `  K
) `  W )
473, 5, 38, 25, 28lspprid1 15853 . . . . . . . . 9  |-  ( ph  ->  E  e.  ( N `
 { E ,  T } ) )
4837, 5, 38, 39, 47lspsnel5a 15852 . . . . . . . 8  |-  ( ph  ->  ( N `  { E } )  C_  ( N `  { E ,  T } ) )
4948, 48unssd 3427 . . . . . . 7  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { E } ) )  C_  ( N `  { E ,  T } ) )
5049sseld 3255 . . . . . 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 32094 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  E >. )
)
531, 16, 13, 46, 11hdmapevec 32097 . . . 4  |-  ( ph  ->  ( S `  E
)  =  ( J `
 E ) )
5452, 53eqtr3d 2392 . . 3  |-  ( ph  ->  ( I `  <. x ,  ( I `  <. E ,  ( J `
 E ) ,  x >. ) ,  E >. )  =  ( J `
 E ) )
553, 5, 38, 25, 28lspprid2 15854 . . . . . . . . 9  |-  ( ph  ->  T  e.  ( N `
 { E ,  T } ) )
5637, 5, 38, 39, 55lspsnel5a 15852 . . . . . . . 8  |-  ( ph  ->  ( N `  { T } )  C_  ( N `  { E ,  T } ) )
5748, 56unssd 3427 . . . . . . 7  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { T } ) )  C_  ( N `  { E ,  T } ) )
5857sseld 3255 . . . . . 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 32094 . . . 4  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. x ,  ( I `  <. E , 
( J `  E
) ,  x >. ) ,  T >. )
)
6160eqcomd 2363 . . 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 32066 . 2  |-  ( ph  ->  ( I `  <. E ,  ( J `  E ) ,  T >. )  =  ( S `
 T ) )
6362eqcomd 2363 1  |-  ( ph  ->  ( S `  T
)  =  ( I `
 <. E ,  ( J `  E ) ,  T >. )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225    u. cun 3226   {csn 3716   {cpr 3717   <.cop 3719   <.cotp 3720    _I cid 4386    |` cres 4773   ` cfv 5337  (class class class)co 5945   Basecbs 13245   0gc0g 13499   -gcsg 14464   LSubSpclss 15788   LSpanclspn 15827   HLchlt 29609   LHypclh 30242   LTrncltrn 30359   DVecHcdvh 31337  LCDualclcd 31845  mapdcmpd 31883  HVMapchvm 32015  HDMap1chdma1 32051  HDMapchdma 32052
This theorem is referenced by:  hdmapval3N  32100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-fal 1320  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-ot 3726  df-uni 3909  df-int 3944  df-iun 3988  df-iin 3989  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-1st 6209  df-2nd 6210  df-tpos 6321  df-undef 6385  df-riota 6391  df-recs 6475  df-rdg 6510  df-1o 6566  df-oadd 6570  df-er 6747  df-map 6862  df-en 6952  df-dom 6953  df-sdom 6954  df-fin 6955  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-n0 10058  df-z 10117  df-uz 10323  df-fz 10875  df-struct 13247  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-ress 13252  df-plusg 13318  df-mulr 13319  df-sca 13321  df-vsca 13322  df-0g 13503  df-mre 13587  df-mrc 13588  df-acs 13590  df-poset 14179  df-plt 14191  df-lub 14207  df-glb 14208  df-join 14209  df-meet 14210  df-p0 14244  df-p1 14245  df-lat 14251  df-clat 14313  df-mnd 14466  df-submnd 14515  df-grp 14588  df-minusg 14589  df-sbg 14590  df-subg 14717  df-cntz 14892  df-oppg 14918  df-lsm 15046  df-cmn 15190  df-abl 15191  df-mgp 15425  df-rng 15439  df-ur 15441  df-oppr 15504  df-dvdsr 15522  df-unit 15523  df-invr 15553  df-dvr 15564  df-drng 15613  df-lmod 15728  df-lss 15789  df-lsp 15828  df-lvec 15955  df-lsatoms 29235  df-lshyp 29236  df-lcv 29278  df-lfl 29317  df-lkr 29345  df-ldual 29383  df-oposet 29435  df-ol 29437  df-oml 29438  df-covers 29525  df-ats 29526  df-atl 29557  df-cvlat 29581  df-hlat 29610  df-llines 29756  df-lplanes 29757  df-lvols 29758  df-lines 29759  df-psubsp 29761  df-pmap 29762  df-padd 30054  df-lhyp 30246  df-laut 30247  df-ldil 30362  df-ltrn 30363  df-trl 30417  df-tgrp 31001  df-tendo 31013  df-edring 31015  df-dveca 31261  df-disoa 31288  df-dvech 31338  df-dib 31398  df-dic 31432  df-dih 31488  df-doch 31607  df-djh 31654  df-lcdual 31846  df-mapd 31884  df-hvmap 32016  df-hdmap1 32053  df-hdmap 32054
  Copyright terms: Public domain W3C validator