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Theorem hdmaplkr 32399
Description: Kernel of the vector to dual map. Line 16 in [Holland95] p. 14. TODO: eliminate  F hypothesis. (Contributed by NM, 9-Jun-2015.)
Hypotheses
Ref Expression
hdmaplkr.h  |-  H  =  ( LHyp `  K
)
hdmaplkr.o  |-  O  =  ( ( ocH `  K
) `  W )
hdmaplkr.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmaplkr.v  |-  V  =  ( Base `  U
)
hdmaplkr.f  |-  F  =  (LFnl `  U )
hdmaplkr.y  |-  Y  =  (LKer `  U )
hdmaplkr.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmaplkr.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmaplkr.x  |-  ( ph  ->  X  e.  V )
Assertion
Ref Expression
hdmaplkr  |-  ( ph  ->  ( Y `  ( S `  X )
)  =  ( O `
 { X }
) )

Proof of Theorem hdmaplkr
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 fveq2 5687 . . . . 5  |-  ( X  =  ( 0g `  U )  ->  ( S `  X )  =  ( S `  ( 0g `  U ) ) )
21fveq2d 5691 . . . 4  |-  ( X  =  ( 0g `  U )  ->  ( Y `  ( S `  X ) )  =  ( Y `  ( S `  ( 0g `  U ) ) ) )
3 sneq 3785 . . . . 5  |-  ( X  =  ( 0g `  U )  ->  { X }  =  { ( 0g `  U ) } )
43fveq2d 5691 . . . 4  |-  ( X  =  ( 0g `  U )  ->  ( O `  { X } )  =  ( O `  { ( 0g `  U ) } ) )
52, 4sseq12d 3337 . . 3  |-  ( X  =  ( 0g `  U )  ->  (
( Y `  ( S `  X )
)  C_  ( O `  { X } )  <-> 
( Y `  ( S `  ( 0g `  U ) ) ) 
C_  ( O `  { ( 0g `  U ) } ) ) )
6 hdmaplkr.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
7 eqid 2404 . . . . . . . . . 10  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
8 hdmaplkr.k . . . . . . . . . 10  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
96, 7, 8lcdlmod 32075 . . . . . . . . 9  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
10 hdmaplkr.u . . . . . . . . . 10  |-  U  =  ( ( DVecH `  K
) `  W )
11 hdmaplkr.v . . . . . . . . . 10  |-  V  =  ( Base `  U
)
12 eqid 2404 . . . . . . . . . 10  |-  ( Base `  ( (LCDual `  K
) `  W )
)  =  ( Base `  ( (LCDual `  K
) `  W )
)
13 hdmaplkr.s . . . . . . . . . 10  |-  S  =  ( (HDMap `  K
) `  W )
14 hdmaplkr.x . . . . . . . . . 10  |-  ( ph  ->  X  e.  V )
156, 10, 11, 7, 12, 13, 8, 14hdmapcl 32316 . . . . . . . . 9  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )
16 eqid 2404 . . . . . . . . . 10  |-  ( LSpan `  ( (LCDual `  K
) `  W )
)  =  ( LSpan `  ( (LCDual `  K
) `  W )
)
1712, 16lspsnid 16024 . . . . . . . . 9  |-  ( ( ( (LCDual `  K
) `  W )  e.  LMod  /\  ( S `  X )  e.  (
Base `  ( (LCDual `  K ) `  W
) ) )  -> 
( S `  X
)  e.  ( (
LSpan `  ( (LCDual `  K ) `  W
) ) `  {
( S `  X
) } ) )
189, 15, 17syl2anc 643 . . . . . . . 8  |-  ( ph  ->  ( S `  X
)  e.  ( (
LSpan `  ( (LCDual `  K ) `  W
) ) `  {
( S `  X
) } ) )
19 eqid 2404 . . . . . . . . . 10  |-  ( LSpan `  U )  =  (
LSpan `  U )
20 eqid 2404 . . . . . . . . . 10  |-  ( (mapd `  K ) `  W
)  =  ( (mapd `  K ) `  W
)
216, 10, 11, 19, 7, 16, 20, 13, 8, 14hdmap10 32326 . . . . . . . . 9  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( ( LSpan `  U ) `  { X } ) )  =  ( ( LSpan `  ( (LCDual `  K
) `  W )
) `  { ( S `  X ) } ) )
22 hdmaplkr.o . . . . . . . . . 10  |-  O  =  ( ( ocH `  K
) `  W )
23 eqid 2404 . . . . . . . . . 10  |-  (LFnl `  U )  =  (LFnl `  U )
24 hdmaplkr.y . . . . . . . . . 10  |-  Y  =  (LKer `  U )
256, 22, 20, 10, 11, 19, 23, 24, 8, 14mapdsn 32124 . . . . . . . . 9  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( ( LSpan `  U ) `  { X } ) )  =  { f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } )
2621, 25eqtr3d 2438 . . . . . . . 8  |-  ( ph  ->  ( ( LSpan `  (
(LCDual `  K ) `  W ) ) `  { ( S `  X ) } )  =  { f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } )
2718, 26eleqtrd 2480 . . . . . . 7  |-  ( ph  ->  ( S `  X
)  e.  { f  e.  (LFnl `  U
)  |  ( O `
 { X }
)  C_  ( Y `  f ) } )
286, 7, 12, 10, 23, 8, 15lcdvbaselfl 32078 . . . . . . . 8  |-  ( ph  ->  ( S `  X
)  e.  (LFnl `  U ) )
29 fveq2 5687 . . . . . . . . . 10  |-  ( f  =  ( S `  X )  ->  ( Y `  f )  =  ( Y `  ( S `  X ) ) )
3029sseq2d 3336 . . . . . . . . 9  |-  ( f  =  ( S `  X )  ->  (
( O `  { X } )  C_  ( Y `  f )  <->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) ) )
3130elrab3 3053 . . . . . . . 8  |-  ( ( S `  X )  e.  (LFnl `  U
)  ->  ( ( S `  X )  e.  { f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } 
<->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) ) )
3228, 31syl 16 . . . . . . 7  |-  ( ph  ->  ( ( S `  X )  e.  {
f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } 
<->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) ) )
3327, 32mpbid 202 . . . . . 6  |-  ( ph  ->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) )
3433adantr 452 . . . . 5  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } ) 
C_  ( Y `  ( S `  X ) ) )
35 eqid 2404 . . . . . 6  |-  (LSHyp `  U )  =  (LSHyp `  U )
366, 10, 8dvhlvec 31592 . . . . . . 7  |-  ( ph  ->  U  e.  LVec )
3736adantr 452 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  U  e.  LVec )
38 eqid 2404 . . . . . . 7  |-  ( 0g
`  U )  =  ( 0g `  U
)
398adantr 452 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4014anim1i 552 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( X  e.  V  /\  X  =/=  ( 0g `  U
) ) )
41 eldifsn 3887 . . . . . . . 8  |-  ( X  e.  ( V  \  { ( 0g `  U ) } )  <-> 
( X  e.  V  /\  X  =/=  ( 0g `  U ) ) )
4240, 41sylibr 204 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  X  e.  ( V  \  { ( 0g `  U ) } ) )
436, 22, 10, 11, 38, 35, 39, 42dochsnshp 31936 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } )  e.  (LSHyp `  U
) )
4428adantr 452 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( S `  X )  e.  (LFnl `  U ) )
45 eqid 2404 . . . . . . . . . . . 12  |-  (Scalar `  U )  =  (Scalar `  U )
46 eqid 2404 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
47 eqid 2404 . . . . . . . . . . . 12  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
486, 10, 11, 45, 46, 7, 47, 8lcd0v 32094 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  (
(LCDual `  K ) `  W ) )  =  ( V  X.  {
( 0g `  (Scalar `  U ) ) } ) )
4948eqeq2d 2415 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  X )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  <->  ( S `  X )  =  ( V  X.  { ( 0g `  (Scalar `  U ) ) } ) ) )
506, 10, 11, 38, 7, 47, 13, 8, 14hdmapeq0 32330 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  X )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  <->  X  =  ( 0g `  U ) ) )
5149, 50bitr3d 247 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  X )  =  ( V  X.  { ( 0g `  (Scalar `  U ) ) } )  <->  X  =  ( 0g `  U ) ) )
5251necon3bid 2602 . . . . . . . 8  |-  ( ph  ->  ( ( S `  X )  =/=  ( V  X.  { ( 0g
`  (Scalar `  U )
) } )  <->  X  =/=  ( 0g `  U ) ) )
5352biimpar 472 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( S `  X )  =/=  ( V  X.  { ( 0g
`  (Scalar `  U )
) } ) )
5411, 45, 46, 35, 23, 24lkrshp 29588 . . . . . . 7  |-  ( ( U  e.  LVec  /\  ( S `  X )  e.  (LFnl `  U )  /\  ( S `  X
)  =/=  ( V  X.  { ( 0g
`  (Scalar `  U )
) } ) )  ->  ( Y `  ( S `  X ) )  e.  (LSHyp `  U ) )
5537, 44, 53, 54syl3anc 1184 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( Y `  ( S `  X
) )  e.  (LSHyp `  U ) )
5635, 37, 43, 55lshpcmp 29471 . . . . 5  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( ( O `  { X } )  C_  ( Y `  ( S `  X ) )  <->  ( O `  { X } )  =  ( Y `  ( S `  X ) ) ) )
5734, 56mpbid 202 . . . 4  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } )  =  ( Y `  ( S `  X ) ) )
58 eqimss2 3361 . . . 4  |-  ( ( O `  { X } )  =  ( Y `  ( S `
 X ) )  ->  ( Y `  ( S `  X ) )  C_  ( O `  { X } ) )
5957, 58syl 16 . . 3  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( Y `  ( S `  X
) )  C_  ( O `  { X } ) )
606, 10, 8dvhlmod 31593 . . . . 5  |-  ( ph  ->  U  e.  LMod )
6111, 38lmod0vcl 15934 . . . . . . . 8  |-  ( U  e.  LMod  ->  ( 0g
`  U )  e.  V )
6260, 61syl 16 . . . . . . 7  |-  ( ph  ->  ( 0g `  U
)  e.  V )
636, 10, 11, 7, 12, 13, 8, 62hdmapcl 32316 . . . . . 6  |-  ( ph  ->  ( S `  ( 0g `  U ) )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
646, 7, 12, 10, 23, 8, 63lcdvbaselfl 32078 . . . . 5  |-  ( ph  ->  ( S `  ( 0g `  U ) )  e.  (LFnl `  U
) )
6511, 23, 24, 60, 64lkrssv 29579 . . . 4  |-  ( ph  ->  ( Y `  ( S `  ( 0g `  U ) ) ) 
C_  V )
666, 10, 22, 11, 38doch0 31841 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O `  {
( 0g `  U
) } )  =  V )
678, 66syl 16 . . . 4  |-  ( ph  ->  ( O `  {
( 0g `  U
) } )  =  V )
6865, 67sseqtr4d 3345 . . 3  |-  ( ph  ->  ( Y `  ( S `  ( 0g `  U ) ) ) 
C_  ( O `  { ( 0g `  U ) } ) )
695, 59, 68pm2.61ne 2642 . 2  |-  ( ph  ->  ( Y `  ( S `  X )
)  C_  ( O `  { X } ) )
7069, 33eqssd 3325 1  |-  ( ph  ->  ( Y `  ( S `  X )
)  =  ( O `
 { X }
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   {crab 2670    \ cdif 3277    C_ wss 3280   {csn 3774    X. cxp 4835   ` cfv 5413   Basecbs 13424  Scalarcsca 13487   0gc0g 13678   LModclmod 15905   LSpanclspn 16002   LVecclvec 16129  LSHypclsh 29458  LFnlclfn 29540  LKerclk 29568   HLchlt 29833   LHypclh 30466   DVecHcdvh 31561   ocHcoch 31830  LCDualclcd 32069  mapdcmpd 32107  HDMapchdma 32276
This theorem is referenced by:  hdmapellkr  32400  hdmapip0  32401  hdmapinvlem1  32404
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-ot 3784  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-tpos 6438  df-undef 6502  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-0g 13682  df-mre 13766  df-mrc 13767  df-acs 13769  df-poset 14358  df-plt 14370  df-lub 14386  df-glb 14387  df-join 14388  df-meet 14389  df-p0 14423  df-p1 14424  df-lat 14430  df-clat 14492  df-mnd 14645  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-subg 14896  df-cntz 15071  df-oppg 15097  df-lsm 15225  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-dvr 15743  df-drng 15792  df-lmod 15907  df-lss 15964  df-lsp 16003  df-lvec 16130  df-lsatoms 29459  df-lshyp 29460  df-lcv 29502  df-lfl 29541  df-lkr 29569  df-ldual 29607  df-oposet 29659  df-ol 29661  df-oml 29662  df-covers 29749  df-ats 29750  df-atl 29781  df-cvlat 29805  df-hlat 29834  df-llines 29980  df-lplanes 29981  df-lvols 29982  df-lines 29983  df-psubsp 29985  df-pmap 29986  df-padd 30278  df-lhyp 30470  df-laut 30471  df-ldil 30586  df-ltrn 30587  df-trl 30641  df-tgrp 31225  df-tendo 31237  df-edring 31239  df-dveca 31485  df-disoa 31512  df-dvech 31562  df-dib 31622  df-dic 31656  df-dih 31712  df-doch 31831  df-djh 31878  df-lcdual 32070  df-mapd 32108  df-hvmap 32240  df-hdmap1 32277  df-hdmap 32278
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