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Theorem nlelshi 27180
Description: The null space of a linear functional is a subspace. (Contributed by NM, 11-Feb-2006.) (Revised by Mario Carneiro, 17-Nov-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
nlelsh.1  |-  T  e. 
LinFn
Assertion
Ref Expression
nlelshi  |-  ( null `  T )  e.  SH

Proof of Theorem nlelshi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-hv0cl 26121 . . 3  |-  0h  e.  ~H
2 nlelsh.1 . . . 4  |-  T  e. 
LinFn
32lnfn0i 27162 . . 3  |-  ( T `
 0h )  =  0
42lnfnfi 27161 . . . 4  |-  T : ~H
--> CC
5 elnlfn 27048 . . . 4  |-  ( T : ~H --> CC  ->  ( 0h  e.  ( null `  T )  <->  ( 0h  e.  ~H  /\  ( T `
 0h )  =  0 ) ) )
64, 5ax-mp 5 . . 3  |-  ( 0h  e.  ( null `  T
)  <->  ( 0h  e.  ~H  /\  ( T `  0h )  =  0
) )
71, 3, 6mpbir2an 918 . 2  |-  0h  e.  ( null `  T )
8 nlfnval 27001 . . . . . . . . . 10  |-  ( T : ~H --> CC  ->  (
null `  T )  =  ( `' T " { 0 } ) )
94, 8ax-mp 5 . . . . . . . . 9  |-  ( null `  T )  =  ( `' T " { 0 } )
10 cnvimass 5345 . . . . . . . . 9  |-  ( `' T " { 0 } )  C_  dom  T
119, 10eqsstri 3519 . . . . . . . 8  |-  ( null `  T )  C_  dom  T
124fdmi 5718 . . . . . . . 8  |-  dom  T  =  ~H
1311, 12sseqtri 3521 . . . . . . 7  |-  ( null `  T )  C_  ~H
1413sseli 3485 . . . . . 6  |-  ( x  e.  ( null `  T
)  ->  x  e.  ~H )
1513sseli 3485 . . . . . 6  |-  ( y  e.  ( null `  T
)  ->  y  e.  ~H )
16 hvaddcl 26130 . . . . . 6  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( x  +h  y
)  e.  ~H )
1714, 15, 16syl2an 475 . . . . 5  |-  ( ( x  e.  ( null `  T )  /\  y  e.  ( null `  T
) )  ->  (
x  +h  y )  e.  ~H )
182lnfnaddi 27163 . . . . . . . 8  |-  ( ( x  e.  ~H  /\  y  e.  ~H )  ->  ( T `  (
x  +h  y ) )  =  ( ( T `  x )  +  ( T `  y ) ) )
1914, 15, 18syl2an 475 . . . . . . 7  |-  ( ( x  e.  ( null `  T )  /\  y  e.  ( null `  T
) )  ->  ( T `  ( x  +h  y ) )  =  ( ( T `  x )  +  ( T `  y ) ) )
20 elnlfn 27048 . . . . . . . . . 10  |-  ( T : ~H --> CC  ->  ( x  e.  ( null `  T )  <->  ( x  e.  ~H  /\  ( T `
 x )  =  0 ) ) )
214, 20ax-mp 5 . . . . . . . . 9  |-  ( x  e.  ( null `  T
)  <->  ( x  e. 
~H  /\  ( T `  x )  =  0 ) )
2221simprbi 462 . . . . . . . 8  |-  ( x  e.  ( null `  T
)  ->  ( T `  x )  =  0 )
23 elnlfn 27048 . . . . . . . . . 10  |-  ( T : ~H --> CC  ->  ( y  e.  ( null `  T )  <->  ( y  e.  ~H  /\  ( T `
 y )  =  0 ) ) )
244, 23ax-mp 5 . . . . . . . . 9  |-  ( y  e.  ( null `  T
)  <->  ( y  e. 
~H  /\  ( T `  y )  =  0 ) )
2524simprbi 462 . . . . . . . 8  |-  ( y  e.  ( null `  T
)  ->  ( T `  y )  =  0 )
2622, 25oveqan12d 6289 . . . . . . 7  |-  ( ( x  e.  ( null `  T )  /\  y  e.  ( null `  T
) )  ->  (
( T `  x
)  +  ( T `
 y ) )  =  ( 0  +  0 ) )
2719, 26eqtrd 2495 . . . . . 6  |-  ( ( x  e.  ( null `  T )  /\  y  e.  ( null `  T
) )  ->  ( T `  ( x  +h  y ) )  =  ( 0  +  0 ) )
28 00id 9744 . . . . . 6  |-  ( 0  +  0 )  =  0
2927, 28syl6eq 2511 . . . . 5  |-  ( ( x  e.  ( null `  T )  /\  y  e.  ( null `  T
) )  ->  ( T `  ( x  +h  y ) )  =  0 )
30 elnlfn 27048 . . . . . 6  |-  ( T : ~H --> CC  ->  ( ( x  +h  y
)  e.  ( null `  T )  <->  ( (
x  +h  y )  e.  ~H  /\  ( T `  ( x  +h  y ) )  =  0 ) ) )
314, 30ax-mp 5 . . . . 5  |-  ( ( x  +h  y )  e.  ( null `  T
)  <->  ( ( x  +h  y )  e. 
~H  /\  ( T `  ( x  +h  y
) )  =  0 ) )
3217, 29, 31sylanbrc 662 . . . 4  |-  ( ( x  e.  ( null `  T )  /\  y  e.  ( null `  T
) )  ->  (
x  +h  y )  e.  ( null `  T
) )
3332rgen2 2879 . . 3  |-  A. x  e.  ( null `  T
) A. y  e.  ( null `  T
) ( x  +h  y )  e.  (
null `  T )
34 hvmulcl 26131 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  ~H )  ->  ( x  .h  y
)  e.  ~H )
3515, 34sylan2 472 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  ( null `  T ) )  -> 
( x  .h  y
)  e.  ~H )
362lnfnmuli 27164 . . . . . . 7  |-  ( ( x  e.  CC  /\  y  e.  ~H )  ->  ( T `  (
x  .h  y ) )  =  ( x  x.  ( T `  y ) ) )
3715, 36sylan2 472 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  ( null `  T ) )  -> 
( T `  (
x  .h  y ) )  =  ( x  x.  ( T `  y ) ) )
3825oveq2d 6286 . . . . . . 7  |-  ( y  e.  ( null `  T
)  ->  ( x  x.  ( T `  y
) )  =  ( x  x.  0 ) )
39 mul01 9748 . . . . . . 7  |-  ( x  e.  CC  ->  (
x  x.  0 )  =  0 )
4038, 39sylan9eqr 2517 . . . . . 6  |-  ( ( x  e.  CC  /\  y  e.  ( null `  T ) )  -> 
( x  x.  ( T `  y )
)  =  0 )
4137, 40eqtrd 2495 . . . . 5  |-  ( ( x  e.  CC  /\  y  e.  ( null `  T ) )  -> 
( T `  (
x  .h  y ) )  =  0 )
42 elnlfn 27048 . . . . . 6  |-  ( T : ~H --> CC  ->  ( ( x  .h  y
)  e.  ( null `  T )  <->  ( (
x  .h  y )  e.  ~H  /\  ( T `  ( x  .h  y ) )  =  0 ) ) )
434, 42ax-mp 5 . . . . 5  |-  ( ( x  .h  y )  e.  ( null `  T
)  <->  ( ( x  .h  y )  e. 
~H  /\  ( T `  ( x  .h  y
) )  =  0 ) )
4435, 41, 43sylanbrc 662 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  ( null `  T ) )  -> 
( x  .h  y
)  e.  ( null `  T ) )
4544rgen2 2879 . . 3  |-  A. x  e.  CC  A. y  e.  ( null `  T
) ( x  .h  y )  e.  (
null `  T )
4633, 45pm3.2i 453 . 2  |-  ( A. x  e.  ( null `  T ) A. y  e.  ( null `  T
) ( x  +h  y )  e.  (
null `  T )  /\  A. x  e.  CC  A. y  e.  ( null `  T ) ( x  .h  y )  e.  ( null `  T
) )
47 issh3 26338 . . 3  |-  ( (
null `  T )  C_ 
~H  ->  ( ( null `  T )  e.  SH  <->  ( 0h  e.  ( null `  T )  /\  ( A. x  e.  ( null `  T ) A. y  e.  ( null `  T ) ( x  +h  y )  e.  ( null `  T
)  /\  A. x  e.  CC  A. y  e.  ( null `  T
) ( x  .h  y )  e.  (
null `  T )
) ) ) )
4813, 47ax-mp 5 . 2  |-  ( (
null `  T )  e.  SH  <->  ( 0h  e.  ( null `  T )  /\  ( A. x  e.  ( null `  T
) A. y  e.  ( null `  T
) ( x  +h  y )  e.  (
null `  T )  /\  A. x  e.  CC  A. y  e.  ( null `  T ) ( x  .h  y )  e.  ( null `  T
) ) ) )
497, 46, 48mpbir2an 918 1  |-  ( null `  T )  e.  SH
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   {csn 4016   `'ccnv 4987   dom cdm 4988   "cima 4991   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481    + caddc 9484    x. cmul 9486   ~Hchil 26037    +h cva 26038    .h csm 26039   0hc0v 26042   SHcsh 26046   nullcnl 26070   LinFnclf 26072
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-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-hilex 26117  ax-hfvadd 26118  ax-hv0cl 26121  ax-hvaddid 26122  ax-hfvmul 26123  ax-hvmulid 26124
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-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  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-er 7303  df-map 7414  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-ltxr 9622  df-sub 9798  df-sh 26325  df-nlfn 26966  df-lnfn 26968
This theorem is referenced by:  nlelchi  27181
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