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Theorem hdmaplkr 37006
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 5871 . . . . 5  |-  ( X  =  ( 0g `  U )  ->  ( S `  X )  =  ( S `  ( 0g `  U ) ) )
21fveq2d 5875 . . . 4  |-  ( X  =  ( 0g `  U )  ->  ( Y `  ( S `  X ) )  =  ( Y `  ( S `  ( 0g `  U ) ) ) )
3 sneq 4042 . . . . 5  |-  ( X  =  ( 0g `  U )  ->  { X }  =  { ( 0g `  U ) } )
43fveq2d 5875 . . . 4  |-  ( X  =  ( 0g `  U )  ->  ( O `  { X } )  =  ( O `  { ( 0g `  U ) } ) )
52, 4sseq12d 3538 . . 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 2467 . . . . . . . . . 10  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
8 hdmaplkr.k . . . . . . . . . 10  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
96, 7, 8lcdlmod 36682 . . . . . . . . 9  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
10 hdmaplkr.u . . . . . . . . . 10  |-  U  =  ( ( DVecH `  K
) `  W )
11 hdmaplkr.v . . . . . . . . . 10  |-  V  =  ( Base `  U
)
12 eqid 2467 . . . . . . . . . 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 36923 . . . . . . . . 9  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )
16 eqid 2467 . . . . . . . . . 10  |-  ( LSpan `  ( (LCDual `  K
) `  W )
)  =  ( LSpan `  ( (LCDual `  K
) `  W )
)
1712, 16lspsnid 17487 . . . . . . . . 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 661 . . . . . . . 8  |-  ( ph  ->  ( S `  X
)  e.  ( (
LSpan `  ( (LCDual `  K ) `  W
) ) `  {
( S `  X
) } ) )
19 eqid 2467 . . . . . . . . . 10  |-  ( LSpan `  U )  =  (
LSpan `  U )
20 eqid 2467 . . . . . . . . . 10  |-  ( (mapd `  K ) `  W
)  =  ( (mapd `  K ) `  W
)
216, 10, 11, 19, 7, 16, 20, 13, 8, 14hdmap10 36933 . . . . . . . . 9  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( ( LSpan `  U ) `  { X } ) )  =  ( ( LSpan `  ( (LCDual `  K
) `  W )
) `  { ( S `  X ) } ) )
22 hdmaplkr.o . . . . . . . . . 10  |-  O  =  ( ( ocH `  K
) `  W )
23 eqid 2467 . . . . . . . . . 10  |-  (LFnl `  U )  =  (LFnl `  U )
24 hdmaplkr.y . . . . . . . . . 10  |-  Y  =  (LKer `  U )
256, 22, 20, 10, 11, 19, 23, 24, 8, 14mapdsn 36731 . . . . . . . . 9  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( ( LSpan `  U ) `  { X } ) )  =  { f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } )
2621, 25eqtr3d 2510 . . . . . . . 8  |-  ( ph  ->  ( ( LSpan `  (
(LCDual `  K ) `  W ) ) `  { ( S `  X ) } )  =  { f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } )
2718, 26eleqtrd 2557 . . . . . . 7  |-  ( ph  ->  ( S `  X
)  e.  { f  e.  (LFnl `  U
)  |  ( O `
 { X }
)  C_  ( Y `  f ) } )
286, 7, 12, 10, 23, 8, 15lcdvbaselfl 36685 . . . . . . . 8  |-  ( ph  ->  ( S `  X
)  e.  (LFnl `  U ) )
29 fveq2 5871 . . . . . . . . . 10  |-  ( f  =  ( S `  X )  ->  ( Y `  f )  =  ( Y `  ( S `  X ) ) )
3029sseq2d 3537 . . . . . . . . 9  |-  ( f  =  ( S `  X )  ->  (
( O `  { X } )  C_  ( Y `  f )  <->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) ) )
3130elrab3 3267 . . . . . . . 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 210 . . . . . 6  |-  ( ph  ->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) )
3433adantr 465 . . . . 5  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } ) 
C_  ( Y `  ( S `  X ) ) )
35 eqid 2467 . . . . . 6  |-  (LSHyp `  U )  =  (LSHyp `  U )
366, 10, 8dvhlvec 36199 . . . . . . 7  |-  ( ph  ->  U  e.  LVec )
3736adantr 465 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  U  e.  LVec )
38 eqid 2467 . . . . . . 7  |-  ( 0g
`  U )  =  ( 0g `  U
)
398adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4014anim1i 568 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( X  e.  V  /\  X  =/=  ( 0g `  U
) ) )
41 eldifsn 4157 . . . . . . . 8  |-  ( X  e.  ( V  \  { ( 0g `  U ) } )  <-> 
( X  e.  V  /\  X  =/=  ( 0g `  U ) ) )
4240, 41sylibr 212 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  X  e.  ( V  \  { ( 0g `  U ) } ) )
436, 22, 10, 11, 38, 35, 39, 42dochsnshp 36543 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } )  e.  (LSHyp `  U
) )
4428adantr 465 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( S `  X )  e.  (LFnl `  U ) )
45 eqid 2467 . . . . . . . . . . . 12  |-  (Scalar `  U )  =  (Scalar `  U )
46 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
47 eqid 2467 . . . . . . . . . . . 12  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
486, 10, 11, 45, 46, 7, 47, 8lcd0v 36701 . . . . . . . . . . 11  |-  ( ph  ->  ( 0g `  (
(LCDual `  K ) `  W ) )  =  ( V  X.  {
( 0g `  (Scalar `  U ) ) } ) )
4948eqeq2d 2481 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  X )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  <->  ( S `  X )  =  ( V  X.  { ( 0g `  (Scalar `  U ) ) } ) ) )
506, 10, 11, 38, 7, 47, 13, 8, 14hdmapeq0 36937 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  X )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  <->  X  =  ( 0g `  U ) ) )
5149, 50bitr3d 255 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  X )  =  ( V  X.  { ( 0g `  (Scalar `  U ) ) } )  <->  X  =  ( 0g `  U ) ) )
5251necon3bid 2725 . . . . . . . 8  |-  ( ph  ->  ( ( S `  X )  =/=  ( V  X.  { ( 0g
`  (Scalar `  U )
) } )  <->  X  =/=  ( 0g `  U ) ) )
5352biimpar 485 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( S `  X )  =/=  ( V  X.  { ( 0g
`  (Scalar `  U )
) } ) )
5411, 45, 46, 35, 23, 24lkrshp 34195 . . . . . . 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 1228 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( Y `  ( S `  X
) )  e.  (LSHyp `  U ) )
5635, 37, 43, 55lshpcmp 34078 . . . . 5  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( ( O `  { X } )  C_  ( Y `  ( S `  X ) )  <->  ( O `  { X } )  =  ( Y `  ( S `  X ) ) ) )
5734, 56mpbid 210 . . . 4  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } )  =  ( Y `  ( S `  X ) ) )
58 eqimss2 3562 . . . 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 36200 . . . . 5  |-  ( ph  ->  U  e.  LMod )
6111, 38lmod0vcl 17389 . . . . . . . 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 36923 . . . . . 6  |-  ( ph  ->  ( S `  ( 0g `  U ) )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
646, 7, 12, 10, 23, 8, 63lcdvbaselfl 36685 . . . . 5  |-  ( ph  ->  ( S `  ( 0g `  U ) )  e.  (LFnl `  U
) )
6511, 23, 24, 60, 64lkrssv 34186 . . . 4  |-  ( ph  ->  ( Y `  ( S `  ( 0g `  U ) ) ) 
C_  V )
666, 10, 22, 11, 38doch0 36448 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O `  {
( 0g `  U
) } )  =  V )
678, 66syl 16 . . . 4  |-  ( ph  ->  ( O `  {
( 0g `  U
) } )  =  V )
6865, 67sseqtr4d 3546 . . 3  |-  ( ph  ->  ( Y `  ( S `  ( 0g `  U ) ) ) 
C_  ( O `  { ( 0g `  U ) } ) )
695, 59, 68pm2.61ne 2782 . 2  |-  ( ph  ->  ( Y `  ( S `  X )
)  C_  ( O `  { X } ) )
7069, 33eqssd 3526 1  |-  ( ph  ->  ( Y `  ( S `  X )
)  =  ( O `
 { X }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2821    \ cdif 3478    C_ wss 3481   {csn 4032    X. cxp 5002   ` cfv 5593   Basecbs 14502  Scalarcsca 14570   0gc0g 14707   LModclmod 17360   LSpanclspn 17465   LVecclvec 17596  LSHypclsh 34065  LFnlclfn 34147  LKerclk 34175   HLchlt 34440   LHypclh 35073   DVecHcdvh 36168   ocHcoch 36437  LCDualclcd 36676  mapdcmpd 36714  HDMapchdma 36883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-riotaBAD 34049
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-ot 4041  df-uni 4251  df-int 4288  df-iun 4332  df-iin 4333  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-tpos 6965  df-undef 7012  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-nn 10547  df-2 10604  df-3 10605  df-4 10606  df-5 10607  df-6 10608  df-n0 10806  df-z 10875  df-uz 11093  df-fz 11683  df-struct 14504  df-ndx 14505  df-slot 14506  df-base 14507  df-sets 14508  df-ress 14509  df-plusg 14580  df-mulr 14581  df-sca 14583  df-vsca 14584  df-0g 14709  df-mre 14853  df-mrc 14854  df-acs 14856  df-poset 15445  df-plt 15457  df-lub 15473  df-glb 15474  df-join 15475  df-meet 15476  df-p0 15538  df-p1 15539  df-lat 15545  df-clat 15607  df-mgm 15741  df-sgrp 15764  df-mnd 15774  df-submnd 15820  df-grp 15906  df-minusg 15907  df-sbg 15908  df-subg 16047  df-cntz 16204  df-oppg 16230  df-lsm 16506  df-cmn 16650  df-abl 16651  df-mgp 16991  df-ur 17003  df-ring 17049  df-oppr 17121  df-dvdsr 17139  df-unit 17140  df-invr 17170  df-dvr 17181  df-drng 17246  df-lmod 17362  df-lss 17427  df-lsp 17466  df-lvec 17597  df-lsatoms 34066  df-lshyp 34067  df-lcv 34109  df-lfl 34148  df-lkr 34176  df-ldual 34214  df-oposet 34266  df-ol 34268  df-oml 34269  df-covers 34356  df-ats 34357  df-atl 34388  df-cvlat 34412  df-hlat 34441  df-llines 34587  df-lplanes 34588  df-lvols 34589  df-lines 34590  df-psubsp 34592  df-pmap 34593  df-padd 34885  df-lhyp 35077  df-laut 35078  df-ldil 35193  df-ltrn 35194  df-trl 35248  df-tgrp 35832  df-tendo 35844  df-edring 35846  df-dveca 36092  df-disoa 36119  df-dvech 36169  df-dib 36229  df-dic 36263  df-dih 36319  df-doch 36438  df-djh 36485  df-lcdual 36677  df-mapd 36715  df-hvmap 36847  df-hdmap1 36884  df-hdmap 36885
This theorem is referenced by:  hdmapellkr  37007  hdmapip0  37008  hdmapinvlem1  37011
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