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Theorem lnon0 21392
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 2566 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  -.  E. x  e.  X  x  =/=  Z )
2 nne 2463 . . . . . 6  |-  ( -.  x  =/=  Z  <->  x  =  Z )
32ralbii 2580 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
41, 3bitr3i 242 . . . 4  |-  ( -. 
E. x  e.  X  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
5 fveq2 5541 . . . . . . . . . 10  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
6 lnon0.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
7 eqid 2296 . . . . . . . . . . 11  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
8 lnon0.6 . . . . . . . . . . 11  |-  Z  =  ( 0vec `  U
)
9 eqid 2296 . . . . . . . . . . 11  |-  ( 0vec `  W )  =  (
0vec `  W )
10 lnon0.7 . . . . . . . . . . 11  |-  L  =  ( U  LnOp  W
)
116, 7, 8, 9, 10lno0 21350 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
125, 11sylan9eqr 2350 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  x  =  Z )  ->  ( T `  x )  =  ( 0vec `  W
) )
1312ex 423 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
x  =  Z  -> 
( T `  x
)  =  ( 0vec `  W ) ) )
1413ralimdv 2635 . . . . . . 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 21349 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
16 ffn 5405 . . . . . . . 8  |-  ( T : X --> ( BaseSet `  W )  ->  T  Fn  X )
1715, 16syl 15 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T  Fn  X )
1814, 17jctild 527 . . . . . 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 5750 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) ) )
20 fvex 5555 . . . . . . . 8  |-  ( 0vec `  W )  e.  _V
2120fconst2 5746 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  T  =  ( X  X.  { ( 0vec `  W ) } ) )
2219, 21bitr3i 242 . . . . . 6  |-  ( ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) )  <->  T  =  ( X  X.  { (
0vec `  W ) } ) )
2318, 22syl6ib 217 . . . . 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 21381 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
26253adant3 975 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
2726eqeq2d 2307 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =  O  <->  T  =  ( X  X.  { (
0vec `  W ) } ) ) )
2823, 27sylibrd 225 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( A. x  e.  X  x  =  Z  ->  T  =  O ) )
294, 28syl5bi 208 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( -.  E. x  e.  X  x  =/=  Z  ->  T  =  O ) )
3029necon1ad 2526 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =/=  O  ->  E. x  e.  X  x  =/=  Z ) )
3130imp 418 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 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557   {csn 3653    X. cxp 4703    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   NrmCVeccnv 21156   BaseSetcba 21158   0veccn0v 21160    LnOp clno 21334    0op c0o 21337
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-sub 9055  df-neg 9056  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-vc 21118  df-nv 21164  df-va 21167  df-ba 21168  df-sm 21169  df-0v 21170  df-nmcv 21172  df-lno 21338  df-0o 21341
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