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Theorem mapd1o 32143
Description: The map defined by df-mapd 32120 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 32080, mapdrval 32142, lclkrs 32034, lclkr 32028,...) to use  T  i^i  ~P C? TODO: maybe get rid of $d's for  g vs.  K U W,. propagate to mapdrn 32144 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 5709 . . . . . 6  |-  (LFnl `  U )  e.  _V
31, 2eqeltri 2482 . . . . 5  |-  F  e. 
_V
43rabex 4322 . . . 4  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) }  e.  _V
5 eqid 2412 . . . 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 5540 . . 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 32122 . . . . 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 16 . . . 4  |-  ( ph  ->  M  =  ( t  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) } ) )
1615fneq1d 5503 . . 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 225 . 2  |-  ( ph  ->  M  Fn  S )
183rabex 4322 . . . . . . 7  |-  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) }  e.  _V
19 eqid 2412 . . . . . . 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 5540 . . . . . 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 32122 . . . . . . . 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 16 . . . . . . 7  |-  ( ph  ->  M  =  ( t  e.  S  |->  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) } ) )
2322fneq1d 5503 . . . . . 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 225 . . . . 5  |-  ( ph  ->  M  Fn  S )
25 fvelrnb 5741 . . . . 5  |-  ( M  Fn  S  ->  (
t  e.  ran  M  <->  E. c  e.  S  ( M `  c )  =  t ) )
2624, 25syl 16 . . . 4  |-  ( ph  ->  ( t  e.  ran  M  <->  E. c  e.  S  ( M `  c )  =  t ) )
277adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  ( K  e.  HL  /\  W  e.  H ) )
28 simpr 448 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  c  e.  S )
298, 9, 10, 1, 11, 12, 13, 27, 28mapdval 32123 . . . . . . . 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 2412 . . . . . . . . . 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 32035 . . . . . . . . 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 3498 . . . . . . . . . 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 4322 . . . . . . . . . . . 12  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  _V
3736elpw 3773 . . . . . . . . . . 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 676 . . . . . . . . . 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 242 . . . . . . . . 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 189 . . . . . . . 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 2486 . . . . . . 7  |-  ( (
ph  /\  c  e.  S )  ->  ( M `  c )  e.  ( T  i^i  ~P C ) )
42 eleq1 2472 . . . . . . 7  |-  ( ( M `  c )  =  t  ->  (
( M `  c
)  e.  ( T  i^i  ~P C )  <-> 
t  e.  ( T  i^i  ~P C ) ) )
4341, 42syl5ibcom 212 . . . . . 6  |-  ( (
ph  /\  c  e.  S )  ->  (
( M `  c
)  =  t  -> 
t  e.  ( T  i^i  ~P C ) ) )
4443rexlimdva 2798 . . . . 5  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  ->  t  e.  ( T  i^i  ~P C
) ) )
45 eqid 2412 . . . . . . . 8  |-  U_ f  e.  t  ( O `  ( L `  f
) )  =  U_ f  e.  t  ( O `  ( L `  f ) )
467adantr 452 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
47 inss1 3529 . . . . . . . . . 10  |-  ( T  i^i  ~P C ) 
C_  T
4847sseli 3312 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  T )
4948adantl 453 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  t  e.  T )
50 inss2 3530 . . . . . . . . . . 11  |-  ( T  i^i  ~P C ) 
C_  ~P C
5150sseli 3312 . . . . . . . . . 10  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  ~P C )
5251elpwid 3776 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  C_  C )
5352adantl 453 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  t  C_  C )
548, 12, 9, 10, 1, 11, 30, 31, 32, 45, 46, 49, 53lcfr 32080 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  U_ f  e.  t  ( O `  ( L `  f
) )  e.  S
)
558, 12, 13, 9, 10, 1, 11, 30, 31, 32, 46, 49, 53, 45mapdrval 32142 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( M `  U_ f  e.  t  ( O `  ( L `  f ) ) )  =  t )
56 fveq2 5695 . . . . . . . . 9  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( M `  c )  =  ( M `  U_ f  e.  t  ( O `  ( L `  f
) ) ) )
5756eqeq1d 2420 . . . . . . . 8  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( ( M `  c )  =  t  <->  ( M `  U_ f  e.  t  ( O `  ( L `
 f ) ) )  =  t ) )
5857rspcev 3020 . . . . . . 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 )
5954, 55, 58syl2anc 643 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  E. c  e.  S  ( M `  c )  =  t )
6059ex 424 . . . . 5  |-  ( ph  ->  ( t  e.  ( T  i^i  ~P C
)  ->  E. c  e.  S  ( M `  c )  =  t ) )
6144, 60impbid 184 . . . 4  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6226, 61bitrd 245 . . 3  |-  ( ph  ->  ( t  e.  ran  M  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6362eqrdv 2410 . 2  |-  ( ph  ->  ran  M  =  ( T  i^i  ~P C
) )
647adantr 452 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
65 simprl 733 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
t  e.  S )
66 simprr 734 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  ->  u  e.  S )
678, 9, 10, 13, 64, 65, 66mapd11 32134 . . . 4  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  <-> 
t  =  u ) )
6867biimpd 199 . . 3  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
6968ralrimivva 2766 . 2  |-  ( ph  ->  A. t  e.  S  A. u  e.  S  ( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
70 dff1o6 5980 . 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 ) ) )
7117, 63, 69, 70syl3anbrc 1138 1  |-  ( ph  ->  M : S -1-1-onto-> ( T  i^i  ~P C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675   {crab 2678   _Vcvv 2924    i^i cin 3287    C_ wss 3288   ~Pcpw 3767   U_ciun 4061    e. cmpt 4234   ran crn 4846    Fn wfn 5416   -1-1-onto->wf1o 5420   ` cfv 5421   LSubSpclss 15971  LFnlclfn 29552  LKerclk 29580  LDualcld 29618   HLchlt 29845   LHypclh 30478   DVecHcdvh 31573   ocHcoch 31842  mapdcmpd 32119
This theorem is referenced by:  mapdrn  32144  mapdcnvcl  32147  mapdcl  32148  mapdcnvid1N  32149  mapdcnvid2  32152
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 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-iin 4064  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-tpos 6446  df-undef 6510  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-4 10024  df-5 10025  df-6 10026  df-n0 10186  df-z 10247  df-uz 10453  df-fz 11008  df-struct 13434  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-ress 13439  df-plusg 13505  df-mulr 13506  df-sca 13508  df-vsca 13509  df-0g 13690  df-mre 13774  df-mrc 13775  df-acs 13777  df-poset 14366  df-plt 14378  df-lub 14394  df-glb 14395  df-join 14396  df-meet 14397  df-p0 14431  df-p1 14432  df-lat 14438  df-clat 14500  df-mnd 14653  df-submnd 14702  df-grp 14775  df-minusg 14776  df-sbg 14777  df-subg 14904  df-cntz 15079  df-oppg 15105  df-lsm 15233  df-cmn 15377  df-abl 15378  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-invr 15740  df-dvr 15751  df-drng 15800  df-lmod 15915  df-lss 15972  df-lsp 16011  df-lvec 16138  df-lsatoms 29471  df-lshyp 29472  df-lcv 29514  df-lfl 29553  df-lkr 29581  df-ldual 29619  df-oposet 29671  df-ol 29673  df-oml 29674  df-covers 29761  df-ats 29762  df-atl 29793  df-cvlat 29817  df-hlat 29846  df-llines 29992  df-lplanes 29993  df-lvols 29994  df-lines 29995  df-psubsp 29997  df-pmap 29998  df-padd 30290  df-lhyp 30482  df-laut 30483  df-ldil 30598  df-ltrn 30599  df-trl 30653  df-tgrp 31237  df-tendo 31249  df-edring 31251  df-dveca 31497  df-disoa 31524  df-dvech 31574  df-dib 31634  df-dic 31668  df-dih 31724  df-doch 31843  df-djh 31890  df-mapd 32120
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