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Theorem hdmap10lem 32654
Description: Lemma for hdmap10 32655. (Contributed by NM, 17-May-2015.)
Hypotheses
Ref Expression
hdmap10.h  |-  H  =  ( LHyp `  K
)
hdmap10.u  |-  U  =  ( ( DVecH `  K
) `  W )
hdmap10.v  |-  V  =  ( Base `  U
)
hdmap10.n  |-  N  =  ( LSpan `  U )
hdmap10.c  |-  C  =  ( (LCDual `  K
) `  W )
hdmap10.l  |-  L  =  ( LSpan `  C )
hdmap10.m  |-  M  =  ( (mapd `  K
) `  W )
hdmap10.s  |-  S  =  ( (HDMap `  K
) `  W )
hdmap10.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
hdmap10.e  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
hdmap10.o  |-  .0.  =  ( 0g `  U )
hdmap10.d  |-  D  =  ( Base `  C
)
hdmap10.j  |-  J  =  ( (HVMap `  K
) `  W )
hdmap10.i  |-  I  =  ( (HDMap1 `  K
) `  W )
hdmap10lem.t  |-  ( ph  ->  T  e.  ( V 
\  {  .0.  }
) )
Assertion
Ref Expression
hdmap10lem  |-  ( ph  ->  ( M `  ( N `  { T } ) )  =  ( L `  {
( S `  T
) } ) )

Proof of Theorem hdmap10lem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 hdmap10.h . . 3  |-  H  =  ( LHyp `  K
)
2 hdmap10.u . . 3  |-  U  =  ( ( DVecH `  K
) `  W )
3 hdmap10.v . . 3  |-  V  =  ( Base `  U
)
4 hdmap10.n . . 3  |-  N  =  ( LSpan `  U )
5 hdmap10.k . . 3  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
6 eqid 2296 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
7 eqid 2296 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
8 hdmap10.o . . . . 5  |-  .0.  =  ( 0g `  U )
9 hdmap10.e . . . . 5  |-  E  = 
<. (  _I  |`  ( Base `  K ) ) ,  (  _I  |`  (
( LTrn `  K ) `  W ) ) >.
101, 6, 7, 2, 3, 8, 9, 5dvheveccl 31924 . . . 4  |-  ( ph  ->  E  e.  ( V 
\  {  .0.  }
) )
11 eldifi 3311 . . . 4  |-  ( E  e.  ( V  \  {  .0.  } )  ->  E  e.  V )
1210, 11syl 15 . . 3  |-  ( ph  ->  E  e.  V )
13 hdmap10lem.t . . . 4  |-  ( ph  ->  T  e.  ( V 
\  {  .0.  }
) )
14 eldifi 3311 . . . 4  |-  ( T  e.  ( V  \  {  .0.  } )  ->  T  e.  V )
1513, 14syl 15 . . 3  |-  ( ph  ->  T  e.  V )
161, 2, 3, 4, 5, 12, 15dvh3dim 32258 . 2  |-  ( ph  ->  E. x  e.  V  -.  x  e.  ( N `  { E ,  T } ) )
17 hdmap10.c . . . . . . 7  |-  C  =  ( (LCDual `  K
) `  W )
18 hdmap10.d . . . . . . 7  |-  D  =  ( Base `  C
)
19 hdmap10.j . . . . . . 7  |-  J  =  ( (HVMap `  K
) `  W )
20 hdmap10.i . . . . . . 7  |-  I  =  ( (HDMap1 `  K
) `  W )
21 hdmap10.s . . . . . . 7  |-  S  =  ( (HDMap `  K
) `  W )
2253ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
23153ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  T  e.  V )
24 simp2 956 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  x  e.  V )
25 eqid 2296 . . . . . . . . . . . 12  |-  ( LSubSp `  U )  =  (
LSubSp `  U )
261, 2, 5dvhlmod 31922 . . . . . . . . . . . 12  |-  ( ph  ->  U  e.  LMod )
273, 25, 4, 26, 12, 15lspprcl 15751 . . . . . . . . . . . 12  |-  ( ph  ->  ( N `  { E ,  T }
)  e.  ( LSubSp `  U ) )
283, 4, 26, 12, 15lspprid1 15770 . . . . . . . . . . . 12  |-  ( ph  ->  E  e.  ( N `
 { E ,  T } ) )
2925, 4, 26, 27, 28lspsnel5a 15769 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { E } )  C_  ( N `  { E ,  T } ) )
303, 4, 26, 12, 15lspprid2 15771 . . . . . . . . . . . 12  |-  ( ph  ->  T  e.  ( N `
 { E ,  T } ) )
3125, 4, 26, 27, 30lspsnel5a 15769 . . . . . . . . . . 11  |-  ( ph  ->  ( N `  { T } )  C_  ( N `  { E ,  T } ) )
3229, 31unssd 3364 . . . . . . . . . 10  |-  ( ph  ->  ( ( N `  { E } )  u.  ( N `  { T } ) )  C_  ( N `  { E ,  T } ) )
3332sseld 3192 . . . . . . . . 9  |-  ( ph  ->  ( x  e.  ( ( N `  { E } )  u.  ( N `  { T } ) )  ->  x  e.  ( N `  { E ,  T } ) ) )
3433con3and 428 . . . . . . . 8  |-  ( (
ph  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) )
35343adant2 974 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  -.  x  e.  ( ( N `  { E } )  u.  ( N `  { T } ) ) )
361, 9, 2, 3, 4, 17, 18, 19, 20, 21, 22, 23, 24, 35hdmapval2 32647 . . . . . 6  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( S `  T )  =  ( I `  <. x ,  ( I `
 <. E ,  ( J `  E ) ,  x >. ) ,  T >. ) )
3736eqcomd 2301 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  (
I `  <. x ,  ( I `  <. E ,  ( J `  E ) ,  x >. ) ,  T >. )  =  ( S `  T ) )
38 eqid 2296 . . . . . 6  |-  ( -g `  U )  =  (
-g `  U )
39 eqid 2296 . . . . . 6  |-  ( -g `  C )  =  (
-g `  C )
40 hdmap10.l . . . . . 6  |-  L  =  ( LSpan `  C )
41 hdmap10.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
42263ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  U  e.  LMod )
43273ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( N `  { E ,  T } )  e.  ( LSubSp `  U )
)
44 simp3 957 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  -.  x  e.  ( N `  { E ,  T } ) )
453, 8, 25, 42, 43, 24, 44lssneln0 15725 . . . . . 6  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  x  e.  ( V  \  {  .0.  } ) )
46 eqid 2296 . . . . . . . . . 10  |-  ( 0g
`  C )  =  ( 0g `  C
)
471, 2, 3, 8, 17, 18, 46, 19, 5, 10hvmapcl2 32578 . . . . . . . . 9  |-  ( ph  ->  ( J `  E
)  e.  ( D 
\  { ( 0g
`  C ) } ) )
48 eldifi 3311 . . . . . . . . 9  |-  ( ( J `  E )  e.  ( D  \  { ( 0g `  C ) } )  ->  ( J `  E )  e.  D
)
4947, 48syl 15 . . . . . . . 8  |-  ( ph  ->  ( J `  E
)  e.  D )
50493ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( J `  E )  e.  D )
511, 2, 3, 8, 4, 17, 40, 41, 19, 5, 10mapdhvmap 32581 . . . . . . . 8  |-  ( ph  ->  ( M `  ( N `  { E } ) )  =  ( L `  {
( J `  E
) } ) )
52513ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( M `  ( N `  { E } ) )  =  ( L `
 { ( J `
 E ) } ) )
531, 2, 5dvhlvec 31921 . . . . . . . . . . 11  |-  ( ph  ->  U  e.  LVec )
54533ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  U  e.  LVec )
55123ad2ant1 976 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  E  e.  V )
563, 4, 54, 24, 55, 23, 44lspindpi 15901 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  (
( N `  {
x } )  =/=  ( N `  { E } )  /\  ( N `  { x } )  =/=  ( N `  { T } ) ) )
5756simpld 445 . . . . . . . 8  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( N `  { x } )  =/=  ( N `  { E } ) )
5857necomd 2542 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( N `  { E } )  =/=  ( N `  { x } ) )
59103ad2ant1 976 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  E  e.  ( V  \  {  .0.  } ) )
601, 2, 3, 8, 4, 17, 18, 40, 41, 20, 22, 50, 52, 58, 59, 24hdmap1cl 32617 . . . . . 6  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  (
I `  <. E , 
( J `  E
) ,  x >. )  e.  D )
61133ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  T  e.  ( V  \  {  .0.  } ) )
621, 2, 3, 17, 18, 21, 5, 15hdmapcl 32645 . . . . . . 7  |-  ( ph  ->  ( S `  T
)  e.  D )
63623ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( S `  T )  e.  D )
6456simprd 449 . . . . . 6  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( N `  { x } )  =/=  ( N `  { T } ) )
65 eqid 2296 . . . . . . . 8  |-  ( I `
 <. E ,  ( J `  E ) ,  x >. )  =  ( I `  <. E ,  ( J `
 E ) ,  x >. )
661, 2, 3, 38, 8, 4, 17, 18, 39, 40, 41, 20, 22, 59, 50, 45, 60, 58, 52hdmap1eq 32614 . . . . . . . 8  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  (
( I `  <. E ,  ( J `  E ) ,  x >. )  =  ( I `
 <. E ,  ( J `  E ) ,  x >. )  <->  ( ( M `  ( N `  { x } ) )  =  ( L `  {
( I `  <. E ,  ( J `  E ) ,  x >. ) } )  /\  ( M `  ( N `
 { ( E ( -g `  U
) x ) } ) )  =  ( L `  { ( ( J `  E
) ( -g `  C
) ( I `  <. E ,  ( J `
 E ) ,  x >. ) ) } ) ) ) )
6765, 66mpbii 202 . . . . . . 7  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  (
( M `  ( N `  { x } ) )  =  ( L `  {
( I `  <. E ,  ( J `  E ) ,  x >. ) } )  /\  ( M `  ( N `
 { ( E ( -g `  U
) x ) } ) )  =  ( L `  { ( ( J `  E
) ( -g `  C
) ( I `  <. E ,  ( J `
 E ) ,  x >. ) ) } ) ) )
6867simpld 445 . . . . . 6  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( M `  ( N `  { x } ) )  =  ( L `
 { ( I `
 <. E ,  ( J `  E ) ,  x >. ) } ) )
691, 2, 3, 38, 8, 4, 17, 18, 39, 40, 41, 20, 22, 45, 60, 61, 63, 64, 68hdmap1eq 32614 . . . . 5  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  (
( I `  <. x ,  ( I `  <. E ,  ( J `
 E ) ,  x >. ) ,  T >. )  =  ( S `
 T )  <->  ( ( M `  ( N `  { T } ) )  =  ( L `
 { ( S `
 T ) } )  /\  ( M `
 ( N `  { ( x (
-g `  U ) T ) } ) )  =  ( L `
 { ( ( I `  <. E , 
( J `  E
) ,  x >. ) ( -g `  C
) ( S `  T ) ) } ) ) ) )
7037, 69mpbid 201 . . . 4  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  (
( M `  ( N `  { T } ) )  =  ( L `  {
( S `  T
) } )  /\  ( M `  ( N `
 { ( x ( -g `  U
) T ) } ) )  =  ( L `  { ( ( I `  <. E ,  ( J `  E ) ,  x >. ) ( -g `  C
) ( S `  T ) ) } ) ) )
7170simpld 445 . . 3  |-  ( (
ph  /\  x  e.  V  /\  -.  x  e.  ( N `  { E ,  T }
) )  ->  ( M `  ( N `  { T } ) )  =  ( L `
 { ( S `
 T ) } ) )
7271rexlimdv3a 2682 . 2  |-  ( ph  ->  ( E. x  e.  V  -.  x  e.  ( N `  { E ,  T }
)  ->  ( M `  ( N `  { T } ) )  =  ( L `  {
( S `  T
) } ) ) )
7316, 72mpd 14 1  |-  ( ph  ->  ( M `  ( N `  { T } ) )  =  ( L `  {
( S `  T
) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   E.wrex 2557    \ cdif 3162    u. cun 3163   {csn 3653   {cpr 3654   <.cop 3656   <.cotp 3657    _I cid 4320    |` cres 4707   ` cfv 5271  (class class class)co 5874   Basecbs 13164   0gc0g 13416   -gcsg 14381   LModclmod 15643   LSubSpclss 15705   LSpanclspn 15744   LVecclvec 15871   HLchlt 30162   LHypclh 30795   LTrncltrn 30912   DVecHcdvh 31890  LCDualclcd 32398  mapdcmpd 32436  HVMapchvm 32568  HDMap1chdma1 32604  HDMapchdma 32605
This theorem is referenced by:  hdmap10  32655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-ot 3663  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-tpos 6250  df-undef 6314  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-sca 13240  df-vsca 13241  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-poset 14096  df-plt 14108  df-lub 14124  df-glb 14125  df-join 14126  df-meet 14127  df-p0 14161  df-p1 14162  df-lat 14168  df-clat 14230  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-sbg 14507  df-subg 14634  df-cntz 14809  df-oppg 14835  df-lsm 14963  df-cmn 15107  df-abl 15108  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-dvr 15481  df-drng 15530  df-lmod 15645  df-lss 15706  df-lsp 15745  df-lvec 15872  df-lsatoms 29788  df-lshyp 29789  df-lcv 29831  df-lfl 29870  df-lkr 29898  df-ldual 29936  df-oposet 29988  df-ol 29990  df-oml 29991  df-covers 30078  df-ats 30079  df-atl 30110  df-cvlat 30134  df-hlat 30163  df-llines 30309  df-lplanes 30310  df-lvols 30311  df-lines 30312  df-psubsp 30314  df-pmap 30315  df-padd 30607  df-lhyp 30799  df-laut 30800  df-ldil 30915  df-ltrn 30916  df-trl 30970  df-tgrp 31554  df-tendo 31566  df-edring 31568  df-dveca 31814  df-disoa 31841  df-dvech 31891  df-dib 31951  df-dic 31985  df-dih 32041  df-doch 32160  df-djh 32207  df-lcdual 32399  df-mapd 32437  df-hvmap 32569  df-hdmap1 32606  df-hdmap 32607
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