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Theorem lshpset2N 33103
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 33102 . . . . 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 487 . . . . . . . . . . 11  |-  ( ( s  e.  H  /\  ( K `  g )  =  s )  -> 
( K `  g
)  e.  H )
76adantll 713 . . . . . . . . . 10  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  ( K `  g )  =  s )  ->  ( K `  g )  e.  H
)
87adantlr 714 . . . . . . . . 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 754 . . . . . . . . . 10  |-  ( ( ( ( W  e. 
LVec  /\  s  e.  H
)  /\  g  e.  F )  /\  ( K `  g )  =  s )  -> 
g  e.  F )
149, 10, 11, 1, 2, 3, 12, 13lkrshp3 33090 . . . . . . . . 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 434 . . . . . . 7  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  g  =/=  ( V  X.  {  .0.  } ) ) )
17 eqimss2 3518 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  s  C_  ( K `  g
) )
18 eqimss 3517 . . . . . . . . 9  |-  ( ( K `  g )  =  s  ->  ( K `  g )  C_  s )
1917, 18eqssd 3482 . . . . . . . 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 533 . . . . . 6  |-  ( ( ( W  e.  LVec  /\  s  e.  H )  /\  g  e.  F
)  ->  ( ( K `  g )  =  s  ->  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
2221reximdva 2934 . . . . 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 434 . . 3  |-  ( W  e.  LVec  ->  ( s  e.  H  ->  E. g  e.  F  ( g  =/=  ( V  X.  {  .0.  } )  /\  s  =  ( K `  g ) ) ) )
259, 10, 11, 1, 2, 3lkrshp 33089 . . . . . . . 8  |-  ( ( W  e.  LVec  /\  g  e.  F  /\  g  =/=  ( V  X.  {  .0.  } ) )  -> 
( K `  g
)  e.  H )
26253adant3r 1216 . . . . . . 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 32963 . . . . . . . 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 1009 . . . . . . 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 2733 . . . . . . . . 9  |-  ( s  =  ( K `  g )  ->  (
s  =/=  V  <->  ( K `  g )  =/=  V
) )
34 uneq1 3612 . . . . . . . . . . . 12  |-  ( s  =  ( K `  g )  ->  (
s  u.  { v } )  =  ( ( K `  g
)  u.  { v } ) )
3534fveq2d 5804 . . . . . . . . . . 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 2868 . . . . . . . . 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 1290 . . . . . . . 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 466 . . . . . . 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 1011 . . . . . 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 2949 . . . 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 32963 . . . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {cab 2439    =/= wne 2648   E.wrex 2800    u. cun 3435   {csn 3986    X. cxp 4947   ` cfv 5527   Basecbs 14293  Scalarcsca 14361   0gc0g 14498   LSubSpclss 17137   LSpanclspn 17176   LVecclvec 17307  LSHypclsh 32959  LFnlclfn 33041  LKerclk 33069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-1st 6688  df-2nd 6689  df-tpos 6856  df-recs 6943  df-rdg 6977  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-nn 10435  df-2 10492  df-3 10493  df-ndx 14296  df-slot 14297  df-base 14298  df-sets 14299  df-ress 14300  df-plusg 14371  df-mulr 14372  df-0g 14500  df-mnd 15535  df-submnd 15585  df-grp 15665  df-minusg 15666  df-sbg 15667  df-subg 15798  df-cntz 15955  df-lsm 16257  df-cmn 16401  df-abl 16402  df-mgp 16715  df-ur 16727  df-rng 16771  df-oppr 16839  df-dvdsr 16857  df-unit 16858  df-invr 16888  df-drng 16958  df-lmod 17074  df-lss 17138  df-lsp 17177  df-lvec 17308  df-lshyp 32961  df-lfl 33042  df-lkr 33070
This theorem is referenced by:  islshpkrN  33104
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