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Theorem mapd1o 31911
Description: The map defined by df-mapd 31888 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 31848, mapdrval 31910, lclkrs 31802, lclkr 31796,...) to use  T  i^i  ~P C? TODO: maybe get rid of $d's for  g vs.  K U W,. propagate to mapdrn 31912 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 5541 . . . . . 6  |-  (LFnl `  U )  e.  _V
31, 2eqeltri 2355 . . . . 5  |-  F  e. 
_V
43rabex 4167 . . . 4  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  t ) }  e.  _V
5 eqid 2285 . . . 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 5374 . . 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 31890 . . . . 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 5337 . . 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 4167 . . . . . . 7  |-  { g  e.  F  |  ( ( O `  ( O `  ( L `  g ) ) )  =  ( L `  g )  /\  ( O `  ( L `  g ) )  C_  t ) }  e.  _V
19 eqid 2285 . . . . . . 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 5374 . . . . . 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 31890 . . . . . . . 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 5337 . . . . . 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 5572 . . . . 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 31891 . . . . . . . 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 2285 . . . . . . . . . 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 31803 . . . . . . . . 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 3360 . . . . . . . . . 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 4167 . . . . . . . . . . . 12  |-  { f  e.  F  |  ( ( O `  ( O `  ( L `  f ) ) )  =  ( L `  f )  /\  ( O `  ( L `  f ) )  C_  c ) }  e.  _V
3736elpw 3633 . . . . . . . . . . 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 2359 . . . . . . 7  |-  ( (
ph  /\  c  e.  S )  ->  ( M `  c )  e.  ( T  i^i  ~P C ) )
42 eleq1 2345 . . . . . . 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 2669 . . . . 5  |-  ( ph  ->  ( E. c  e.  S  ( M `  c )  =  t  ->  t  e.  ( T  i^i  ~P C
) ) )
45 eqid 2285 . . . . . . . 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 3391 . . . . . . . . . 10  |-  ( T  i^i  ~P C ) 
C_  T
4847sseli 3178 . . . . . . . . 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 3392 . . . . . . . . . . 11  |-  ( T  i^i  ~P C ) 
C_  ~P C
5150sseli 3178 . . . . . . . . . 10  |-  ( t  e.  ( T  i^i  ~P C )  ->  t  e.  ~P C )
52 elpwi 3635 . . . . . . . . . 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 31848 . . . . . . 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 31910 . . . . . . 7  |-  ( (
ph  /\  t  e.  ( T  i^i  ~P C
) )  ->  ( M `  U_ f  e.  t  ( O `  ( L `  f ) ) )  =  t )
57 fveq2 5527 . . . . . . . . 9  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( M `  c )  =  ( M `  U_ f  e.  t  ( O `  ( L `  f
) ) ) )
5857eqeq1d 2293 . . . . . . . 8  |-  ( c  =  U_ f  e.  t  ( O `  ( L `  f ) )  ->  ( ( M `  c )  =  t  <->  ( M `  U_ f  e.  t  ( O `  ( L `
 f ) ) )  =  t ) )
5958rspcev 2886 . . . . . . 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 2283 . 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 31902 . . . 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 2637 . 2  |-  ( ph  ->  A. t  e.  S  A. u  e.  S  ( ( M `  t )  =  ( M `  u )  ->  t  =  u ) )
71 dff1o6 5793 . 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 1625    e. wcel 1686   A.wral 2545   E.wrex 2546   {crab 2549   _Vcvv 2790    i^i cin 3153    C_ wss 3154   ~Pcpw 3627   U_ciun 3907    e. cmpt 4079   ran crn 4692    Fn wfn 5252   -1-1-onto->wf1o 5256   ` cfv 5257   LSubSpclss 15691  LFnlclfn 29320  LKerclk 29348  LDualcld 29386   HLchlt 29613   LHypclh 30246   DVecHcdvh 31341   ocHcoch 31610  mapdcmpd 31887
This theorem is referenced by:  mapdrn  31912  mapdcnvcl  31915  mapdcl  31916  mapdcnvid1N  31917  mapdcnvid2  31920
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-1st 6124  df-2nd 6125  df-tpos 6236  df-undef 6300  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-er 6662  df-map 6776  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-n0 9968  df-z 10027  df-uz 10233  df-fz 10785  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-sca 13226  df-vsca 13227  df-0g 13406  df-mre 13490  df-mrc 13491  df-acs 13493  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-mnd 14369  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-subg 14620  df-cntz 14795  df-oppg 14821  df-lsm 14949  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-dvr 15467  df-drng 15516  df-lmod 15631  df-lss 15692  df-lsp 15731  df-lvec 15858  df-lsatoms 29239  df-lshyp 29240  df-lcv 29282  df-lfl 29321  df-lkr 29349  df-ldual 29387  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-tgrp 31005  df-tendo 31017  df-edring 31019  df-dveca 31265  df-disoa 31292  df-dvech 31342  df-dib 31402  df-dic 31436  df-dih 31492  df-doch 31611  df-djh 31658  df-mapd 31888
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