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Theorem lnon0 21376
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 2553 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  -.  E. x  e.  X  x  =/=  Z )
2 nne 2450 . . . . . 6  |-  ( -.  x  =/=  Z  <->  x  =  Z )
32ralbii 2567 . . . . 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 5525 . . . . . . . . . 10  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
6 lnon0.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
7 eqid 2283 . . . . . . . . . . 11  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
8 lnon0.6 . . . . . . . . . . 11  |-  Z  =  ( 0vec `  U
)
9 eqid 2283 . . . . . . . . . . 11  |-  ( 0vec `  W )  =  (
0vec `  W )
10 lnon0.7 . . . . . . . . . . 11  |-  L  =  ( U  LnOp  W
)
116, 7, 8, 9, 10lno0 21334 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
125, 11sylan9eqr 2337 . . . . . . . . 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 2622 . . . . . . 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 21333 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
16 ffn 5389 . . . . . . . 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 5734 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) ) )
20 fvex 5539 . . . . . . . 8  |-  ( 0vec `  W )  e.  _V
2120fconst2 5730 . . . . . . 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 21365 . . . . . . 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 2294 . . . . 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 2513 . 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 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {csn 3640    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   NrmCVeccnv 21140   BaseSetcba 21142   0veccn0v 21144    LnOp clno 21318    0op c0o 21321
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-sub 9039  df-neg 9040  df-grpo 20858  df-gid 20859  df-ginv 20860  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-lno 21322  df-0o 21325
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