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Theorem mapd1o 35389
Description: The map defined by df-mapd 35366 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 35326, mapdrval 35388, lclkrs 35280, lclkr 35274,...) to use  T  i^i  ~P C? TODO: maybe get rid of $d's for  g vs.  K U W,. propagate to mapdrn 35390 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 5722 . . . . . 6  |-  (LFnl `  U )  e.  _V
31, 2eqeltri 2513 . . . . 5  |-  F  e. 
_V
43rabex 4464 . . . 4  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) }  e.  _V
5 eqid 2443 . . . 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 5560 . . 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 35368 . . . . 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 5522 . . 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 233 . 2  |-  ( ph  ->  M  Fn  S )
183rabex 4464 . . . . . . 7  |-  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) }  e.  _V
19 eqid 2443 . . . . . . 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 5560 . . . . . 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 35368 . . . . . . . 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 5522 . . . . . 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 233 . . . . 5  |-  ( ph  ->  M  Fn  S )
25 fvelrnb 5760 . . . . 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 465 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  ( K  e.  HL  /\  W  e.  H ) )
28 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  c  e.  S )
298, 9, 10, 1, 11, 12, 13, 27, 28mapdval 35369 . . . . . . . 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 2443 . . . . . . . . . 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 35281 . . . . . . . . 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 3560 . . . . . . . . . 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 4464 . . . . . . . . . . . 12  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  _V
3736elpw 3887 . . . . . . . . . . 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 694 . . . . . . . . . 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 250 . . . . . . . . 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 196 . . . . . . . 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 2517 . . . . . . 7  |-  ( (
ph  /\  c  e.  S )  ->  ( M `  c )  e.  ( T  i^i  ~P C ) )
42 eleq1 2503 . . . . . . 7  |-  ( ( M `  c )  =  t  ->  (
( M `  c
)  e.  ( T  i^i  ~P C )  <-> 
t  e.  ( T  i^i  ~P C ) ) )
4341, 42syl5ibcom 220 . . . . . 6  |-  ( (
ph  /\  c  e.  S )  ->  (
( M `  c
)  =  t  -> 
t  e.  ( T  i^i  ~P C ) ) )
4443rexlimdva 2862 . . . . 5  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  ->  t  e.  ( T  i^i  ~P C
) ) )
45 eqid 2443 . . . . . . . 8  |-  U_ f  e.  t  ( O `  ( L `  f
) )  =  U_ f  e.  t  ( O `  ( L `  f ) )
467adantr 465 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
47 inss1 3591 . . . . . . . . . 10  |-  ( T  i^i  ~P C ) 
C_  T
4847sseli 3373 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  T )
4948adantl 466 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  t  e.  T )
50 inss2 3592 . . . . . . . . . . 11  |-  ( T  i^i  ~P C ) 
C_  ~P C
5150sseli 3373 . . . . . . . . . 10  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  ~P C )
5251elpwid 3891 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  C_  C )
5352adantl 466 . . . . . . . 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 35326 . . . . . . 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 35388 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( M `  U_ f  e.  t  ( O `  ( L `  f ) ) )  =  t )
56 fveq2 5712 . . . . . . . . 9  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( M `  c )  =  ( M `  U_ f  e.  t  ( O `  ( L `  f
) ) ) )
5756eqeq1d 2451 . . . . . . . 8  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( ( M `  c )  =  t  <->  ( M `  U_ f  e.  t  ( O `  ( L `
 f ) ) )  =  t ) )
5857rspcev 3094 . . . . . . 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 661 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  E. c  e.  S  ( M `  c )  =  t )
6059ex 434 . . . . 5  |-  ( ph  ->  ( t  e.  ( T  i^i  ~P C
)  ->  E. c  e.  S  ( M `  c )  =  t ) )
6144, 60impbid 191 . . . 4  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6226, 61bitrd 253 . . 3  |-  ( ph  ->  ( t  e.  ran  M  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6362eqrdv 2441 . 2  |-  ( ph  ->  ran  M  =  ( T  i^i  ~P C
) )
647adantr 465 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
65 simprl 755 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
t  e.  S )
66 simprr 756 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  ->  u  e.  S )
678, 9, 10, 13, 64, 65, 66mapd11 35380 . . . 4  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  <-> 
t  =  u ) )
6867biimpd 207 . . 3  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
6968ralrimivva 2829 . 2  |-  ( ph  ->  A. t  e.  S  A. u  e.  S  ( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
70 dff1o6 6003 . 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 1172 1  |-  ( ph  ->  M : S -1-1-onto-> ( T  i^i  ~P C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2736   E.wrex 2737   {crab 2740   _Vcvv 2993    i^i cin 3348    C_ wss 3349   ~Pcpw 3881   U_ciun 4192    e. cmpt 4371   ran crn 4862    Fn wfn 5434   -1-1-onto->wf1o 5438   ` cfv 5439   LSubSpclss 17035  LFnlclfn 32798  LKerclk 32826  LDualcld 32864   HLchlt 33091   LHypclh 33724   DVecHcdvh 34819   ocHcoch 35088  mapdcmpd 35365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380  ax-riotaBAD 32700
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-pss 3365  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-tp 3903  df-op 3905  df-uni 4113  df-int 4150  df-iun 4194  df-iin 4195  df-br 4314  df-opab 4372  df-mpt 4373  df-tr 4407  df-eprel 4653  df-id 4657  df-po 4662  df-so 4663  df-fr 4700  df-we 4702  df-ord 4743  df-on 4744  df-lim 4745  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-of 6341  df-om 6498  df-1st 6598  df-2nd 6599  df-tpos 6766  df-undef 6813  df-recs 6853  df-rdg 6887  df-1o 6941  df-oadd 6945  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-fin 7335  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-nn 10344  df-2 10401  df-3 10402  df-4 10403  df-5 10404  df-6 10405  df-n0 10601  df-z 10668  df-uz 10883  df-fz 11459  df-struct 14197  df-ndx 14198  df-slot 14199  df-base 14200  df-sets 14201  df-ress 14202  df-plusg 14272  df-mulr 14273  df-sca 14275  df-vsca 14276  df-0g 14401  df-mre 14545  df-mrc 14546  df-acs 14548  df-poset 15137  df-plt 15149  df-lub 15165  df-glb 15166  df-join 15167  df-meet 15168  df-p0 15230  df-p1 15231  df-lat 15237  df-clat 15299  df-mnd 15436  df-submnd 15486  df-grp 15566  df-minusg 15567  df-sbg 15568  df-subg 15699  df-cntz 15856  df-oppg 15882  df-lsm 16156  df-cmn 16300  df-abl 16301  df-mgp 16614  df-ur 16626  df-rng 16669  df-oppr 16737  df-dvdsr 16755  df-unit 16756  df-invr 16786  df-dvr 16797  df-drng 16856  df-lmod 16972  df-lss 17036  df-lsp 17075  df-lvec 17206  df-lsatoms 32717  df-lshyp 32718  df-lcv 32760  df-lfl 32799  df-lkr 32827  df-ldual 32865  df-oposet 32917  df-ol 32919  df-oml 32920  df-covers 33007  df-ats 33008  df-atl 33039  df-cvlat 33063  df-hlat 33092  df-llines 33238  df-lplanes 33239  df-lvols 33240  df-lines 33241  df-psubsp 33243  df-pmap 33244  df-padd 33536  df-lhyp 33728  df-laut 33729  df-ldil 33844  df-ltrn 33845  df-trl 33899  df-tgrp 34483  df-tendo 34495  df-edring 34497  df-dveca 34743  df-disoa 34770  df-dvech 34820  df-dib 34880  df-dic 34914  df-dih 34970  df-doch 35089  df-djh 35136  df-mapd 35366
This theorem is referenced by:  mapdrn  35390  mapdcnvcl  35393  mapdcl  35394  mapdcnvid1N  35395  mapdcnvid2  35398
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