MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lnon0 Unicode version

Theorem lnon0 22260
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 2684 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  -.  E. x  e.  X  x  =/=  Z )
2 nne 2579 . . . . . 6  |-  ( -.  x  =/=  Z  <->  x  =  Z )
32ralbii 2698 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
41, 3bitr3i 243 . . . 4  |-  ( -. 
E. x  e.  X  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
5 fveq2 5695 . . . . . . . . . 10  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
6 lnon0.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
7 eqid 2412 . . . . . . . . . . 11  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
8 lnon0.6 . . . . . . . . . . 11  |-  Z  =  ( 0vec `  U
)
9 eqid 2412 . . . . . . . . . . 11  |-  ( 0vec `  W )  =  (
0vec `  W )
10 lnon0.7 . . . . . . . . . . 11  |-  L  =  ( U  LnOp  W
)
116, 7, 8, 9, 10lno0 22218 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
125, 11sylan9eqr 2466 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  x  =  Z )  ->  ( T `  x )  =  ( 0vec `  W
) )
1312ex 424 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
x  =  Z  -> 
( T `  x
)  =  ( 0vec `  W ) ) )
1413ralimdv 2753 . . . . . . 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 22217 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
16 ffn 5558 . . . . . . . 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 528 . . . . . 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 5921 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) ) )
20 fvex 5709 . . . . . . . 8  |-  ( 0vec `  W )  e.  _V
2120fconst2 5915 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  T  =  ( X  X.  { ( 0vec `  W ) } ) )
2219, 21bitr3i 243 . . . . . 6  |-  ( ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) )  <->  T  =  ( X  X.  { (
0vec `  W ) } ) )
2318, 22syl6ib 218 . . . . 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 22249 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
26253adant3 977 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
2726eqeq2d 2423 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =  O  <->  T  =  ( X  X.  { (
0vec `  W ) } ) ) )
2823, 27sylibrd 226 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( A. x  e.  X  x  =  Z  ->  T  =  O ) )
294, 28syl5bi 209 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( -.  E. x  e.  X  x  =/=  Z  ->  T  =  O ) )
3029necon1ad 2642 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =/=  O  ->  E. x  e.  X  x  =/=  Z ) )
3130imp 419 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   E.wrex 2675   {csn 3782    X. cxp 4843    Fn wfn 5416   -->wf 5417   ` cfv 5421  (class class class)co 6048   NrmCVeccnv 22024   BaseSetcba 22026   0veccn0v 22028    LnOp clno 22202    0op c0o 22205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-po 4471  df-so 4472  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-er 6872  df-map 6987  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-ltxr 9089  df-sub 9257  df-neg 9258  df-grpo 21740  df-gid 21741  df-ginv 21742  df-ablo 21831  df-vc 21986  df-nv 22032  df-va 22035  df-ba 22036  df-sm 22037  df-0v 22038  df-nmcv 22040  df-lno 22206  df-0o 22209
  Copyright terms: Public domain W3C validator