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Theorem lshpset2N 35241
Description: The set of all hyperplanes of a left module or left vector space equals the set of all kernels of nonzero functionals. (Contributed by NM, 17-Jul-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lshpset2.v  |-  V  =  ( Base `  W
)
lshpset2.d  |-  D  =  (Scalar `  W )
lshpset2.z  |-  .0.  =  ( 0g `  D )
lshpset2.h  |-  H  =  (LSHyp `  W )
lshpset2.f  |-  F  =  (LFnl `  W )
lshpset2.k  |-  K  =  (LKer `  W )
Assertion
Ref Expression
lshpset2N  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
Distinct variable groups:    g, F    g, s, H    g, K    g, V    g, W, s
Allowed substitution hints:    D( g, s)    F( s)    K( s)    V( s)    .0. ( g, s)

Proof of Theorem lshpset2N
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 lshpset2.h . . . . . 6  |-  H  =  (LSHyp `  W )
2 lshpset2.f . . . . . 6  |-  F  =  (LFnl `  W )
3 lshpset2.k . . . . . 6  |-  K  =  (LKer `  W )
41, 2, 3lshpkrex 35240 . . . . 5  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  E. g  e.  F  ( K `  g )  =  s )
5 eleq1 2526 . . . . . . . . . . . 12  |-  ( ( K `  g )  =  s  ->  (
( K `  g
)  e.  H  <->  s  e.  H ) )
65biimparc 485 . . . . . . . . . . 11  |-  ( ( s  e.  H  /\  ( K `  g )  =  s )  -> 
( K `  g
)  e.  H )
76adantll 711 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  ( K `  g )  =  s )  ->  ( K `  g )  e.  H
)
87adantlr 712 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
( K `  g
)  e.  H )
9 lshpset2.v . . . . . . . . . 10  |-  V  =  ( Base `  W
)
10 lshpset2.d . . . . . . . . . 10  |-  D  =  (Scalar `  W )
11 lshpset2.z . . . . . . . . . 10  |-  .0.  =  ( 0g `  D )
12 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  ->  W  e.  LVec )
13 simplr 753 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
g  e.  F )
149, 10, 11, 1, 2, 3, 12, 13lkrshp3 35228 . . . . . . . . 9  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
( ( K `  g )  e.  H  <->  g  =/=  ( V  X.  {  .0.  } ) ) )
158, 14mpbid 210 . . . . . . . 8  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
g  =/=  ( V  X.  {  .0.  }
) )
1615ex 432 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  g  =/=  ( V  X.  {  .0.  } ) ) )
17 eqimss2 3542 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  s  C_  ( K `  g
) )
18 eqimss 3541 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  ( K `  g )  C_  s )
1917, 18eqssd 3506 . . . . . . . 8  |-  ( ( K `  g )  =  s  ->  s  =  ( K `  g ) )
2019a1i 11 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  s  =  ( K `  g
) ) )
2116, 20jcad 531 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
2221reximdva 2929 . . . . 5  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  ( E. g  e.  F  ( K `  g )  =  s  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
234, 22mpd 15 . . . 4  |-  ( ( W  e.  LVec  /\  s  e.  H )  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) )
2423ex 432 . . 3  |-  ( W  e.  LVec  ->  ( s  e.  H  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
259, 10, 11, 1, 2, 3lkrshp 35227 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  g
)  e.  H )
26253adant3r 1223 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  ( K `  g )  e.  H )
27 eqid 2454 . . . . . . . . 9  |-  ( LSpan `  W )  =  (
LSpan `  W )
28 eqid 2454 . . . . . . . . 9  |-  ( LSubSp `  W )  =  (
LSubSp `  W )
299, 27, 28, 1islshp 35101 . . . . . . . 8  |-  ( W  e.  LVec  ->  ( ( K `  g )  e.  H  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
30293ad2ant1 1015 . . . . . . 7  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
( K `  g
)  e.  H  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
3126, 30mpbid 210 . . . . . 6  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
( K `  g
)  e.  ( LSubSp `  W )  /\  ( K `  g )  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
32 eleq1 2526 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  e.  ( LSubSp `  W )  <->  ( K `  g )  e.  (
LSubSp `  W ) ) )
33 neeq1 2735 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  =/=  V  <->  ( K `  g )  =/=  V
) )
34 uneq1 3637 . . . . . . . . . . . 12  |-  ( s  =  ( K `  g )  ->  (
s  u.  { v } )  =  ( ( K `  g
)  u.  { v } ) )
3534fveq2d 5852 . . . . . . . . . . 11  |-  ( s  =  ( K `  g )  ->  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  ( ( LSpan `  W ) `  (
( K `  g
)  u.  { v } ) ) )
3635eqeq1d 2456 . . . . . . . . . 10  |-  ( s  =  ( K `  g )  ->  (
( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V  <->  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
3736rexbidv 2965 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  ( E. v  e.  V  ( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V  <->  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) )
3832, 33, 373anbi123d 1297 . . . . . . . 8  |-  ( s  =  ( K `  g )  ->  (
( s  e.  (
LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V )  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
3938adantl 464 . . . . . . 7  |-  ( ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  ->  ( (
s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V )  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
40393ad2ant3 1017 . . . . . 6  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
( s  e.  (
LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  (
( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V )  <->  ( ( K `  g )  e.  ( LSubSp `  W )  /\  ( K `  g
)  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( ( K `  g )  u.  {
v } ) )  =  V ) ) )
4131, 40mpbird 232 . . . . 5  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) )  ->  (
s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W ) `  ( s  u.  {
v } ) )  =  V ) )
4241rexlimdv3a 2948 . . . 4  |-  ( W  e.  LVec  ->  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  ->  ( s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V ) ) )
439, 27, 28, 1islshp 35101 . . . 4  |-  ( W  e.  LVec  ->  ( s  e.  H  <->  ( s  e.  ( LSubSp `  W )  /\  s  =/=  V  /\  E. v  e.  V  ( ( LSpan `  W
) `  ( s  u.  { v } ) )  =  V ) ) )
4442, 43sylibrd 234 . . 3  |-  ( W  e.  LVec  ->  ( E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) )  ->  s  e.  H ) )
4524, 44impbid 191 . 2  |-  ( W  e.  LVec  ->  ( s  e.  H  <->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
4645abbi2dv 2591 1  |-  ( W  e.  LVec  ->  H  =  { s  |  E. g  e.  F  (
g  =/=  ( V  X.  {  .0.  }
)  /\  s  =  ( K `  g ) ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   {cab 2439    =/= wne 2649   E.wrex 2805    u. cun 3459   {csn 4016    X. cxp 4986   ` cfv 5570   Basecbs 14716  Scalarcsca 14787   0gc0g 14929   LSubSpclss 17773   LSpanclspn 17812   LVecclvec 17943  LSHypclsh 35097  LFnlclfn 35179  LKerclk 35207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-grp 16256  df-minusg 16257  df-sbg 16258  df-subg 16397  df-cntz 16554  df-lsm 16855  df-cmn 16999  df-abl 17000  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-drng 17593  df-lmod 17709  df-lss 17774  df-lsp 17813  df-lvec 17944  df-lshyp 35099  df-lfl 35180  df-lkr 35208
This theorem is referenced by:  islshpkrN  35242
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