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Theorem lnon0 22330
Description: The domain of a nonzero linear operator contains a nonzero vector. (Contributed by NM, 15-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
lnon0.1  |-  X  =  ( BaseSet `  U )
lnon0.6  |-  Z  =  ( 0vec `  U
)
lnon0.0  |-  O  =  ( U  0op  W
)
lnon0.7  |-  L  =  ( U  LnOp  W
)
Assertion
Ref Expression
lnon0  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  T  =/=  O )  ->  E. x  e.  X  x  =/=  Z )
Distinct variable groups:    x, L    x, T    x, U    x, W    x, X
Allowed substitution hints:    O( x)    Z( x)

Proof of Theorem lnon0
StepHypRef Expression
1 ralnex 2721 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  -.  E. x  e.  X  x  =/=  Z )
2 nne 2611 . . . . . 6  |-  ( -.  x  =/=  Z  <->  x  =  Z )
32ralbii 2735 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
41, 3bitr3i 244 . . . 4  |-  ( -. 
E. x  e.  X  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
5 fveq2 5757 . . . . . . . . . 10  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
6 lnon0.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
7 eqid 2442 . . . . . . . . . . 11  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
8 lnon0.6 . . . . . . . . . . 11  |-  Z  =  ( 0vec `  U
)
9 eqid 2442 . . . . . . . . . . 11  |-  ( 0vec `  W )  =  (
0vec `  W )
10 lnon0.7 . . . . . . . . . . 11  |-  L  =  ( U  LnOp  W
)
116, 7, 8, 9, 10lno0 22288 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
125, 11sylan9eqr 2496 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  x  =  Z )  ->  ( T `  x )  =  ( 0vec `  W
) )
1312ex 425 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
x  =  Z  -> 
( T `  x
)  =  ( 0vec `  W ) ) )
1413ralimdv 2791 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( A. x  e.  X  x  =  Z  ->  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) ) )
156, 7, 10lnof 22287 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
16 ffn 5620 . . . . . . . 8  |-  ( T : X --> ( BaseSet `  W )  ->  T  Fn  X )
1715, 16syl 16 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T  Fn  X )
1814, 17jctild 529 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( A. x  e.  X  x  =  Z  ->  ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) ) ) )
19 fconstfv 5983 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) ) )
20 fvex 5771 . . . . . . . 8  |-  ( 0vec `  W )  e.  _V
2120fconst2 5977 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  T  =  ( X  X.  { ( 0vec `  W ) } ) )
2219, 21bitr3i 244 . . . . . 6  |-  ( ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) )  <->  T  =  ( X  X.  { (
0vec `  W ) } ) )
2318, 22syl6ib 219 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( A. x  e.  X  x  =  Z  ->  T  =  ( X  X.  { ( 0vec `  W
) } ) ) )
24 lnon0.0 . . . . . . . 8  |-  O  =  ( U  0op  W
)
256, 9, 240ofval 22319 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
26253adant3 978 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
2726eqeq2d 2453 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =  O  <->  T  =  ( X  X.  { (
0vec `  W ) } ) ) )
2823, 27sylibrd 227 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( A. x  e.  X  x  =  Z  ->  T  =  O ) )
294, 28syl5bi 210 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( -.  E. x  e.  X  x  =/=  Z  ->  T  =  O ) )
3029necon1ad 2677 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =/=  O  ->  E. x  e.  X  x  =/=  Z ) )
3130imp 420 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  T  =/=  O )  ->  E. x  e.  X  x  =/=  Z )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   E.wrex 2712   {csn 3838    X. cxp 4905    Fn wfn 5478   -->wf 5479   ` cfv 5483  (class class class)co 6110   NrmCVeccnv 22094   BaseSetcba 22096   0veccn0v 22098    LnOp clno 22272    0op c0o 22275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-ltxr 9156  df-sub 9324  df-neg 9325  df-grpo 21810  df-gid 21811  df-ginv 21812  df-ablo 21901  df-vc 22056  df-nv 22102  df-va 22105  df-ba 22106  df-sm 22107  df-0v 22108  df-nmcv 22110  df-lno 22276  df-0o 22279
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