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Theorem hdmaplkr 35555
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 5879 . . . . 5  |-  ( X  =  ( 0g `  U )  ->  ( S `  X )  =  ( S `  ( 0g `  U ) ) )
21fveq2d 5883 . . . 4  |-  ( X  =  ( 0g `  U )  ->  ( Y `  ( S `  X ) )  =  ( Y `  ( S `  ( 0g `  U ) ) ) )
3 sneq 3969 . . . . 5  |-  ( X  =  ( 0g `  U )  ->  { X }  =  { ( 0g `  U ) } )
43fveq2d 5883 . . . 4  |-  ( X  =  ( 0g `  U )  ->  ( O `  { X } )  =  ( O `  { ( 0g `  U ) } ) )
52, 4sseq12d 3447 . . 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 2471 . . . . . . . . . 10  |-  ( (LCDual `  K ) `  W
)  =  ( (LCDual `  K ) `  W
)
8 hdmaplkr.k . . . . . . . . . 10  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
96, 7, 8lcdlmod 35231 . . . . . . . . 9  |-  ( ph  ->  ( (LCDual `  K
) `  W )  e.  LMod )
10 hdmaplkr.u . . . . . . . . . 10  |-  U  =  ( ( DVecH `  K
) `  W )
11 hdmaplkr.v . . . . . . . . . 10  |-  V  =  ( Base `  U
)
12 eqid 2471 . . . . . . . . . 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 35472 . . . . . . . . 9  |-  ( ph  ->  ( S `  X
)  e.  ( Base `  ( (LCDual `  K
) `  W )
) )
16 eqid 2471 . . . . . . . . . 10  |-  ( LSpan `  ( (LCDual `  K
) `  W )
)  =  ( LSpan `  ( (LCDual `  K
) `  W )
)
1712, 16lspsnid 18294 . . . . . . . . 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 673 . . . . . . . 8  |-  ( ph  ->  ( S `  X
)  e.  ( (
LSpan `  ( (LCDual `  K ) `  W
) ) `  {
( S `  X
) } ) )
19 eqid 2471 . . . . . . . . . 10  |-  ( LSpan `  U )  =  (
LSpan `  U )
20 eqid 2471 . . . . . . . . . 10  |-  ( (mapd `  K ) `  W
)  =  ( (mapd `  K ) `  W
)
216, 10, 11, 19, 7, 16, 20, 13, 8, 14hdmap10 35482 . . . . . . . . 9  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( ( LSpan `  U ) `  { X } ) )  =  ( ( LSpan `  ( (LCDual `  K
) `  W )
) `  { ( S `  X ) } ) )
22 hdmaplkr.o . . . . . . . . . 10  |-  O  =  ( ( ocH `  K
) `  W )
23 eqid 2471 . . . . . . . . . 10  |-  (LFnl `  U )  =  (LFnl `  U )
24 hdmaplkr.y . . . . . . . . . 10  |-  Y  =  (LKer `  U )
256, 22, 20, 10, 11, 19, 23, 24, 8, 14mapdsn 35280 . . . . . . . . 9  |-  ( ph  ->  ( ( (mapd `  K ) `  W
) `  ( ( LSpan `  U ) `  { X } ) )  =  { f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } )
2621, 25eqtr3d 2507 . . . . . . . 8  |-  ( ph  ->  ( ( LSpan `  (
(LCDual `  K ) `  W ) ) `  { ( S `  X ) } )  =  { f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } )
2718, 26eleqtrd 2551 . . . . . . 7  |-  ( ph  ->  ( S `  X
)  e.  { f  e.  (LFnl `  U
)  |  ( O `
 { X }
)  C_  ( Y `  f ) } )
286, 7, 12, 10, 23, 8, 15lcdvbaselfl 35234 . . . . . . . 8  |-  ( ph  ->  ( S `  X
)  e.  (LFnl `  U ) )
29 fveq2 5879 . . . . . . . . . 10  |-  ( f  =  ( S `  X )  ->  ( Y `  f )  =  ( Y `  ( S `  X ) ) )
3029sseq2d 3446 . . . . . . . . 9  |-  ( f  =  ( S `  X )  ->  (
( O `  { X } )  C_  ( Y `  f )  <->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) ) )
3130elrab3 3185 . . . . . . . 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 17 . . . . . . 7  |-  ( ph  ->  ( ( S `  X )  e.  {
f  e.  (LFnl `  U )  |  ( O `  { X } )  C_  ( Y `  f ) } 
<->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) ) )
3327, 32mpbid 215 . . . . . 6  |-  ( ph  ->  ( O `  { X } )  C_  ( Y `  ( S `  X ) ) )
3433adantr 472 . . . . 5  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } ) 
C_  ( Y `  ( S `  X ) ) )
35 eqid 2471 . . . . . 6  |-  (LSHyp `  U )  =  (LSHyp `  U )
366, 10, 8dvhlvec 34748 . . . . . . 7  |-  ( ph  ->  U  e.  LVec )
3736adantr 472 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  U  e.  LVec )
38 eqid 2471 . . . . . . 7  |-  ( 0g
`  U )  =  ( 0g `  U
)
398adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( K  e.  HL  /\  W  e.  H ) )
4014anim1i 578 . . . . . . . 8  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( X  e.  V  /\  X  =/=  ( 0g `  U
) ) )
41 eldifsn 4088 . . . . . . . 8  |-  ( X  e.  ( V  \  { ( 0g `  U ) } )  <-> 
( X  e.  V  /\  X  =/=  ( 0g `  U ) ) )
4240, 41sylibr 217 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  X  e.  ( V  \  { ( 0g `  U ) } ) )
436, 22, 10, 11, 38, 35, 39, 42dochsnshp 35092 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } )  e.  (LSHyp `  U
) )
4428adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( S `  X )  e.  (LFnl `  U ) )
45 eqid 2471 . . . . . . . . . . . 12  |-  (Scalar `  U )  =  (Scalar `  U )
46 eqid 2471 . . . . . . . . . . . 12  |-  ( 0g
`  (Scalar `  U )
)  =  ( 0g
`  (Scalar `  U )
)
47 eqid 2471 . . . . . . . . . . . 12  |-  ( 0g
`  ( (LCDual `  K ) `  W
) )  =  ( 0g `  ( (LCDual `  K ) `  W
) )
486, 10, 11, 45, 46, 7, 47, 8lcd0v 35250 . . . . . . . . . . 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 35486 . . . . . . . . . 10  |-  ( ph  ->  ( ( S `  X )  =  ( 0g `  ( (LCDual `  K ) `  W
) )  <->  X  =  ( 0g `  U ) ) )
5149, 50bitr3d 263 . . . . . . . . 9  |-  ( ph  ->  ( ( S `  X )  =  ( V  X.  { ( 0g `  (Scalar `  U ) ) } )  <->  X  =  ( 0g `  U ) ) )
5251necon3bid 2687 . . . . . . . 8  |-  ( ph  ->  ( ( S `  X )  =/=  ( V  X.  { ( 0g
`  (Scalar `  U )
) } )  <->  X  =/=  ( 0g `  U ) ) )
5352biimpar 493 . . . . . . 7  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( S `  X )  =/=  ( V  X.  { ( 0g
`  (Scalar `  U )
) } ) )
5411, 45, 46, 35, 23, 24lkrshp 32742 . . . . . . 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 1292 . . . . . 6  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( Y `  ( S `  X
) )  e.  (LSHyp `  U ) )
5635, 37, 43, 55lshpcmp 32625 . . . . 5  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( ( O `  { X } )  C_  ( Y `  ( S `  X ) )  <->  ( O `  { X } )  =  ( Y `  ( S `  X ) ) ) )
5734, 56mpbid 215 . . . 4  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( O `  { X } )  =  ( Y `  ( S `  X ) ) )
58 eqimss2 3471 . . . 4  |-  ( ( O `  { X } )  =  ( Y `  ( S `
 X ) )  ->  ( Y `  ( S `  X ) )  C_  ( O `  { X } ) )
5957, 58syl 17 . . 3  |-  ( (
ph  /\  X  =/=  ( 0g `  U ) )  ->  ( Y `  ( S `  X
) )  C_  ( O `  { X } ) )
606, 10, 8dvhlmod 34749 . . . . 5  |-  ( ph  ->  U  e.  LMod )
6111, 38lmod0vcl 18198 . . . . . . . 8  |-  ( U  e.  LMod  ->  ( 0g
`  U )  e.  V )
6260, 61syl 17 . . . . . . 7  |-  ( ph  ->  ( 0g `  U
)  e.  V )
636, 10, 11, 7, 12, 13, 8, 62hdmapcl 35472 . . . . . 6  |-  ( ph  ->  ( S `  ( 0g `  U ) )  e.  ( Base `  (
(LCDual `  K ) `  W ) ) )
646, 7, 12, 10, 23, 8, 63lcdvbaselfl 35234 . . . . 5  |-  ( ph  ->  ( S `  ( 0g `  U ) )  e.  (LFnl `  U
) )
6511, 23, 24, 60, 64lkrssv 32733 . . . 4  |-  ( ph  ->  ( Y `  ( S `  ( 0g `  U ) ) ) 
C_  V )
666, 10, 22, 11, 38doch0 34997 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( O `  {
( 0g `  U
) } )  =  V )
678, 66syl 17 . . . 4  |-  ( ph  ->  ( O `  {
( 0g `  U
) } )  =  V )
6865, 67sseqtr4d 3455 . . 3  |-  ( ph  ->  ( Y `  ( S `  ( 0g `  U ) ) ) 
C_  ( O `  { ( 0g `  U ) } ) )
695, 59, 68pm2.61ne 2728 . 2  |-  ( ph  ->  ( Y `  ( S `  X )
)  C_  ( O `  { X } ) )
7069, 33eqssd 3435 1  |-  ( ph  ->  ( Y `  ( S `  X )
)  =  ( O `
 { X }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {crab 2760    \ cdif 3387    C_ wss 3390   {csn 3959    X. cxp 4837   ` cfv 5589   Basecbs 15199  Scalarcsca 15271   0gc0g 15416   LModclmod 18169   LSpanclspn 18272   LVecclvec 18403  LSHypclsh 32612  LFnlclfn 32694  LKerclk 32722   HLchlt 32987   LHypclh 33620   DVecHcdvh 34717   ocHcoch 34986  LCDualclcd 35225  mapdcmpd 35263  HDMapchdma 35432
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-riotaBAD 32589
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-ot 3968  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-tpos 6991  df-undef 7038  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-sca 15284  df-vsca 15285  df-0g 15418  df-mre 15570  df-mrc 15571  df-acs 15573  df-preset 16251  df-poset 16269  df-plt 16282  df-lub 16298  df-glb 16299  df-join 16300  df-meet 16301  df-p0 16363  df-p1 16364  df-lat 16370  df-clat 16432  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-grp 16751  df-minusg 16752  df-sbg 16753  df-subg 16892  df-cntz 17049  df-oppg 17075  df-lsm 17366  df-cmn 17510  df-abl 17511  df-mgp 17802  df-ur 17814  df-ring 17860  df-oppr 17929  df-dvdsr 17947  df-unit 17948  df-invr 17978  df-dvr 17989  df-drng 18055  df-lmod 18171  df-lss 18234  df-lsp 18273  df-lvec 18404  df-lsatoms 32613  df-lshyp 32614  df-lcv 32656  df-lfl 32695  df-lkr 32723  df-ldual 32761  df-oposet 32813  df-ol 32815  df-oml 32816  df-covers 32903  df-ats 32904  df-atl 32935  df-cvlat 32959  df-hlat 32988  df-llines 33134  df-lplanes 33135  df-lvols 33136  df-lines 33137  df-psubsp 33139  df-pmap 33140  df-padd 33432  df-lhyp 33624  df-laut 33625  df-ldil 33740  df-ltrn 33741  df-trl 33796  df-tgrp 34381  df-tendo 34393  df-edring 34395  df-dveca 34641  df-disoa 34668  df-dvech 34718  df-dib 34778  df-dic 34812  df-dih 34868  df-doch 34987  df-djh 35034  df-lcdual 35226  df-mapd 35264  df-hvmap 35396  df-hdmap1 35433  df-hdmap 35434
This theorem is referenced by:  hdmapellkr  35556  hdmapip0  35557  hdmapinvlem1  35560
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