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Theorem mapd1o 31838
Description: The map defined by df-mapd 31815 is one-to-one and onto the set of dual subspaces of functionals with closed kernels. This shows  M satisfies part of the definition of projectivity of [Baer] p. 40. TODO: change theorems leading to this (lcfr 31775, mapdrval 31837, lclkrs 31729, lclkr 31723,...) to use  T  i^i  ~P C? TODO: maybe get rid of $d's for  g vs.  K U W,. propagate to mapdrn 31839 and any others. (Contributed by NM, 12-Mar-2015.)
Hypotheses
Ref Expression
mapd1o.h  |-  H  =  ( LHyp `  K
)
mapd1o.o  |-  O  =  ( ( ocH `  K
) `  W )
mapd1o.m  |-  M  =  ( (mapd `  K
) `  W )
mapd1o.u  |-  U  =  ( ( DVecH `  K
) `  W )
mapd1o.s  |-  S  =  ( LSubSp `  U )
mapd1o.f  |-  F  =  (LFnl `  U )
mapd1o.l  |-  L  =  (LKer `  U )
mapd1o.d  |-  D  =  (LDual `  U )
mapd1o.t  |-  T  =  ( LSubSp `  D )
mapd1o.c  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
mapd1o.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
Assertion
Ref Expression
mapd1o  |-  ( ph  ->  M : S -1-1-onto-> ( T  i^i  ~P C ) )
Distinct variable groups:    g, F    g, K    g, L    g, O    U, g    g, W
Allowed substitution hints:    ph( g)    C( g)    D( g)    S( g)    T( g)    H( g)    M( g)

Proof of Theorem mapd1o
Dummy variables  f 
c  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mapd1o.f . . . . . 6  |-  F  =  (LFnl `  U )
2 fvex 5539 . . . . . 6  |-  (LFnl `  U )  e.  _V
31, 2eqeltri 2353 . . . . 5  |-  F  e. 
_V
43rabex 4165 . . . 4  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) }  e.  _V
5 eqid 2283 . . . 4  |-  ( t  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) } )  =  ( t  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) } )
64, 5fnmpti 5372 . . 3  |-  ( t  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) } )  Fn  S
7 mapd1o.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
8 mapd1o.h . . . . . 6  |-  H  =  ( LHyp `  K
)
9 mapd1o.u . . . . . 6  |-  U  =  ( ( DVecH `  K
) `  W )
10 mapd1o.s . . . . . 6  |-  S  =  ( LSubSp `  U )
11 mapd1o.l . . . . . 6  |-  L  =  (LKer `  U )
12 mapd1o.o . . . . . 6  |-  O  =  ( ( ocH `  K
) `  W )
13 mapd1o.m . . . . . 6  |-  M  =  ( (mapd `  K
) `  W )
148, 9, 10, 1, 11, 12, 13mapdfval 31817 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  M  =  ( t  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) } ) )
157, 14syl 15 . . . 4  |-  ( ph  ->  M  =  ( t  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) } ) )
1615fneq1d 5335 . . 3  |-  ( ph  ->  ( M  Fn  S  <->  ( t  e.  S  |->  { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  t ) } )  Fn  S ) )
176, 16mpbiri 224 . 2  |-  ( ph  ->  M  Fn  S )
183rabex 4165 . . . . . . 7  |-  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) }  e.  _V
19 eqid 2283 . . . . . . 7  |-  ( t  e.  S  |->  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) } )  =  ( t  e.  S  |->  { g  e.  F  |  ( ( O `  ( O `
 ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) } )
2018, 19fnmpti 5372 . . . . . 6  |-  ( t  e.  S  |->  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) } )  Fn  S
218, 9, 10, 1, 11, 12, 13mapdfval 31817 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  M  =  ( t  e.  S  |->  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) } ) )
227, 21syl 15 . . . . . . 7  |-  ( ph  ->  M  =  ( t  e.  S  |->  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) } ) )
2322fneq1d 5335 . . . . . 6  |-  ( ph  ->  ( M  Fn  S  <->  ( t  e.  S  |->  { g  e.  F  | 
( ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g )  /\  ( O `  ( L `
 g ) ) 
C_  t ) } )  Fn  S ) )
2420, 23mpbiri 224 . . . . 5  |-  ( ph  ->  M  Fn  S )
25 fvelrnb 5570 . . . . 5  |-  ( M  Fn  S  ->  (
t  e.  ran  M  <->  E. c  e.  S  ( M `  c )  =  t ) )
2624, 25syl 15 . . . 4  |-  ( ph  ->  ( t  e.  ran  M  <->  E. c  e.  S  ( M `  c )  =  t ) )
277adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  ( K  e.  HL  /\  W  e.  H ) )
28 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  c  e.  S )
298, 9, 10, 1, 11, 12, 13, 27, 28mapdval 31818 . . . . . . . 8  |-  ( (
ph  /\  c  e.  S )  ->  ( M `  c )  =  { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) } )
30 mapd1o.d . . . . . . . . . 10  |-  D  =  (LDual `  U )
31 mapd1o.t . . . . . . . . . 10  |-  T  =  ( LSubSp `  D )
32 mapd1o.c . . . . . . . . . 10  |-  C  =  { g  e.  F  |  ( O `  ( O `  ( L `
 g ) ) )  =  ( L `
 g ) }
33 eqid 2283 . . . . . . . . . 10  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  =  { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }
348, 12, 9, 10, 1, 11, 30, 31, 32, 33, 27, 28lclkrs2 31730 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  ( { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  T  /\  { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  c ) } 
C_  C ) )
35 elin 3358 . . . . . . . . . 10  |-  ( { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  c ) }  e.  ( T  i^i  ~P C )  <->  ( {
f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  c ) }  e.  T  /\  {
f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  c ) }  e.  ~P C ) )
363rabex 4165 . . . . . . . . . . . 12  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  _V
3736elpw 3631 . . . . . . . . . . 11  |-  ( { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  c ) }  e.  ~P C  <->  { f  e.  F  |  (
( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  C_  C )
3837anbi2i 675 . . . . . . . . . 10  |-  ( ( { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  T  /\  { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  c ) }  e.  ~P C )  <-> 
( { f  e.  F  |  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  T  /\  { f  e.  F  |  ( ( O `  ( O `
 ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  C_  C ) )
3935, 38bitr2i 241 . . . . . . . . 9  |-  ( ( { f  e.  F  |  ( ( O `
 ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  T  /\  { f  e.  F  | 
( ( O `  ( O `  ( L `
 f ) ) )  =  ( L `
 f )  /\  ( O `  ( L `
 f ) ) 
C_  c ) } 
C_  C )  <->  { f  e.  F  |  (
( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  ( T  i^i  ~P C
) )
4034, 39sylib 188 . . . . . . . 8  |-  ( (
ph  /\  c  e.  S )  ->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  ( T  i^i  ~P C
) )
4129, 40eqeltrd 2357 . . . . . . 7  |-  ( (
ph  /\  c  e.  S )  ->  ( M `  c )  e.  ( T  i^i  ~P C ) )
42 eleq1 2343 . . . . . . 7  |-  ( ( M `  c )  =  t  ->  (
( M `  c
)  e.  ( T  i^i  ~P C )  <-> 
t  e.  ( T  i^i  ~P C ) ) )
4341, 42syl5ibcom 211 . . . . . 6  |-  ( (
ph  /\  c  e.  S )  ->  (
( M `  c
)  =  t  -> 
t  e.  ( T  i^i  ~P C ) ) )
4443rexlimdva 2667 . . . . 5  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  ->  t  e.  ( T  i^i  ~P C
) ) )
45 eqid 2283 . . . . . . . 8  |-  U_ f  e.  t  ( O `  ( L `  f
) )  =  U_ f  e.  t  ( O `  ( L `  f ) )
467adantr 451 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
47 inss1 3389 . . . . . . . . . 10  |-  ( T  i^i  ~P C ) 
C_  T
4847sseli 3176 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  T )
4948adantl 452 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  t  e.  T )
50 inss2 3390 . . . . . . . . . . 11  |-  ( T  i^i  ~P C ) 
C_  ~P C
5150sseli 3176 . . . . . . . . . 10  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  ~P C )
52 elpwi 3633 . . . . . . . . . 10  |-  ( t  e.  ~P C  -> 
t  C_  C )
5351, 52syl 15 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  C_  C )
5453adantl 452 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  t  C_  C )
558, 12, 9, 10, 1, 11, 30, 31, 32, 45, 46, 49, 54lcfr 31775 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  U_ f  e.  t  ( O `  ( L `  f
) )  e.  S
)
568, 12, 13, 9, 10, 1, 11, 30, 31, 32, 46, 49, 54, 45mapdrval 31837 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( M `  U_ f  e.  t  ( O `  ( L `  f ) ) )  =  t )
57 fveq2 5525 . . . . . . . . 9  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( M `  c )  =  ( M `  U_ f  e.  t  ( O `  ( L `  f
) ) ) )
5857eqeq1d 2291 . . . . . . . 8  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( ( M `  c )  =  t  <->  ( M `  U_ f  e.  t  ( O `  ( L `
 f ) ) )  =  t ) )
5958rspcev 2884 . . . . . . 7  |-  ( (
U_ f  e.  t  ( O `  ( L `  f )
)  e.  S  /\  ( M `  U_ f  e.  t  ( O `  ( L `  f
) ) )  =  t )  ->  E. c  e.  S  ( M `  c )  =  t )
6055, 56, 59syl2anc 642 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  E. c  e.  S  ( M `  c )  =  t )
6160ex 423 . . . . 5  |-  ( ph  ->  ( t  e.  ( T  i^i  ~P C
)  ->  E. c  e.  S  ( M `  c )  =  t ) )
6244, 61impbid 183 . . . 4  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6326, 62bitrd 244 . . 3  |-  ( ph  ->  ( t  e.  ran  M  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6463eqrdv 2281 . 2  |-  ( ph  ->  ran  M  =  ( T  i^i  ~P C
) )
657adantr 451 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
66 simprl 732 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
t  e.  S )
67 simprr 733 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  ->  u  e.  S )
688, 9, 10, 13, 65, 66, 67mapd11 31829 . . . 4  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  <-> 
t  =  u ) )
6968biimpd 198 . . 3  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
7069ralrimivva 2635 . 2  |-  ( ph  ->  A. t  e.  S  A. u  e.  S  ( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
71 dff1o6 5791 . 2  |-  ( M : S -1-1-onto-> ( T  i^i  ~P C )  <->  ( M  Fn  S  /\  ran  M  =  ( T  i^i  ~P C )  /\  A. t  e.  S  A. u  e.  S  (
( M `  t
)  =  ( M `
 u )  -> 
t  =  u ) ) )
7217, 64, 70, 71syl3anbrc 1136 1  |-  ( ph  ->  M : S -1-1-onto-> ( T  i^i  ~P C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U_ciun 3905    e. cmpt 4077   ran crn 4690    Fn wfn 5250   -1-1-onto->wf1o 5254   ` cfv 5255   LSubSpclss 15689  LFnlclfn 29247  LKerclk 29275  LDualcld 29313   HLchlt 29540   LHypclh 30173   DVecHcdvh 31268   ocHcoch 31537  mapdcmpd 31814
This theorem is referenced by:  mapdrn  31839  mapdcnvcl  31842  mapdcl  31843  mapdcnvid1N  31844  mapdcnvid2  31847
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-oppg 14819  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-lsatoms 29166  df-lshyp 29167  df-lcv 29209  df-lfl 29248  df-lkr 29276  df-ldual 29314  df-oposet 29366  df-ol 29368  df-oml 29369  df-covers 29456  df-ats 29457  df-atl 29488  df-cvlat 29512  df-hlat 29541  df-llines 29687  df-lplanes 29688  df-lvols 29689  df-lines 29690  df-psubsp 29692  df-pmap 29693  df-padd 29985  df-lhyp 30177  df-laut 30178  df-ldil 30293  df-ltrn 30294  df-trl 30348  df-tgrp 30932  df-tendo 30944  df-edring 30946  df-dveca 31192  df-disoa 31219  df-dvech 31269  df-dib 31329  df-dic 31363  df-dih 31419  df-doch 31538  df-djh 31585  df-mapd 31815
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