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Theorem mapd1o 35287
Description: The map defined by df-mapd 35264 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 35224, mapdrval 35286, lclkrs 35178, lclkr 35172,...) to use  T  i^i  ~P C? TODO: maybe get rid of $d's for  g vs.  K U W,. propagate to mapdrn 35288 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 5889 . . . . . 6  |-  (LFnl `  U )  e.  _V
31, 2eqeltri 2545 . . . . 5  |-  F  e. 
_V
43rabex 4550 . . . 4  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) }  e.  _V
5 eqid 2471 . . . 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 5716 . . 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 35266 . . . . 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 17 . . . 4  |-  ( ph  ->  M  =  ( t  e.  S  |->  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) } ) )
1615fneq1d 5676 . . 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 241 . 2  |-  ( ph  ->  M  Fn  S )
183rabex 4550 . . . . . . 7  |-  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) }  e.  _V
19 eqid 2471 . . . . . . 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 5716 . . . . . 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 35266 . . . . . . . 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 17 . . . . . . 7  |-  ( ph  ->  M  =  ( t  e.  S  |->  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) } ) )
2322fneq1d 5676 . . . . . 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 241 . . . . 5  |-  ( ph  ->  M  Fn  S )
25 fvelrnb 5926 . . . . 5  |-  ( M  Fn  S  ->  (
t  e.  ran  M  <->  E. c  e.  S  ( M `  c )  =  t ) )
2624, 25syl 17 . . . 4  |-  ( ph  ->  ( t  e.  ran  M  <->  E. c  e.  S  ( M `  c )  =  t ) )
277adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  ( K  e.  HL  /\  W  e.  H ) )
28 simpr 468 . . . . . . . . 9  |-  ( (
ph  /\  c  e.  S )  ->  c  e.  S )
298, 9, 10, 1, 11, 12, 13, 27, 28mapdval 35267 . . . . . . . 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 2471 . . . . . . . . . 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 35179 . . . . . . . . 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 3608 . . . . . . . . . 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 4550 . . . . . . . . . . . 12  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  _V
3736elpw 3948 . . . . . . . . . . 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 708 . . . . . . . . . 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 258 . . . . . . . . 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 201 . . . . . . . 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 2549 . . . . . . 7  |-  ( (
ph  /\  c  e.  S )  ->  ( M `  c )  e.  ( T  i^i  ~P C ) )
42 eleq1 2537 . . . . . . 7  |-  ( ( M `  c )  =  t  ->  (
( M `  c
)  e.  ( T  i^i  ~P C )  <-> 
t  e.  ( T  i^i  ~P C ) ) )
4341, 42syl5ibcom 228 . . . . . 6  |-  ( (
ph  /\  c  e.  S )  ->  (
( M `  c
)  =  t  -> 
t  e.  ( T  i^i  ~P C ) ) )
4443rexlimdva 2871 . . . . 5  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  ->  t  e.  ( T  i^i  ~P C
) ) )
45 eqid 2471 . . . . . . . 8  |-  U_ f  e.  t  ( O `  ( L `  f
) )  =  U_ f  e.  t  ( O `  ( L `  f ) )
467adantr 472 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( K  e.  HL  /\  W  e.  H ) )
47 inss1 3643 . . . . . . . . . 10  |-  ( T  i^i  ~P C ) 
C_  T
4847sseli 3414 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  T )
4948adantl 473 . . . . . . . 8  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  t  e.  T )
50 inss2 3644 . . . . . . . . . . 11  |-  ( T  i^i  ~P C ) 
C_  ~P C
5150sseli 3414 . . . . . . . . . 10  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  ~P C )
5251elpwid 3952 . . . . . . . . 9  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  C_  C )
5352adantl 473 . . . . . . . 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 35224 . . . . . . 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 35286 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( M `  U_ f  e.  t  ( O `  ( L `  f ) ) )  =  t )
56 fveq2 5879 . . . . . . . . 9  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( M `  c )  =  ( M `  U_ f  e.  t  ( O `  ( L `  f
) ) ) )
5756eqeq1d 2473 . . . . . . . 8  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( ( M `  c )  =  t  <->  ( M `  U_ f  e.  t  ( O `  ( L `
 f ) ) )  =  t ) )
5857rspcev 3136 . . . . . . 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 673 . . . . . 6  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  E. c  e.  S  ( M `  c )  =  t )
6059ex 441 . . . . 5  |-  ( ph  ->  ( t  e.  ( T  i^i  ~P C
)  ->  E. c  e.  S  ( M `  c )  =  t ) )
6144, 60impbid 195 . . . 4  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6226, 61bitrd 261 . . 3  |-  ( ph  ->  ( t  e.  ran  M  <-> 
t  e.  ( T  i^i  ~P C ) ) )
6362eqrdv 2469 . 2  |-  ( ph  ->  ran  M  =  ( T  i^i  ~P C
) )
647adantr 472 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( K  e.  HL  /\  W  e.  H ) )
65 simprl 772 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
t  e.  S )
66 simprr 774 . . . . 5  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  ->  u  e.  S )
678, 9, 10, 13, 64, 65, 66mapd11 35278 . . . 4  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  <-> 
t  =  u ) )
6867biimpd 212 . . 3  |-  ( (
ph  /\  ( t  e.  S  /\  u  e.  S ) )  -> 
( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
6968ralrimivva 2814 . 2  |-  ( ph  ->  A. t  e.  S  A. u  e.  S  ( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
70 dff1o6 6192 . 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 1214 1  |-  ( ph  ->  M : S -1-1-onto-> ( T  i^i  ~P C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   E.wrex 2757   {crab 2760   _Vcvv 3031    i^i cin 3389    C_ wss 3390   ~Pcpw 3942   U_ciun 4269    |-> cmpt 4454   ran crn 4840    Fn wfn 5584   -1-1-onto->wf1o 5588   ` cfv 5589   LSubSpclss 18233  LFnlclfn 32694  LKerclk 32722  LDualcld 32760   HLchlt 32987   LHypclh 33620   DVecHcdvh 34717   ocHcoch 34986  mapdcmpd 35263
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-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-mapd 35264
This theorem is referenced by:  mapdrn  35288  mapdcnvcl  35291  mapdcl  35292  mapdcnvid1N  35293  mapdcnvid2  35296
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