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Theorem lnon0 24196
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 2723 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  -.  E. x  e.  X  x  =/=  Z )
2 nne 2610 . . . . . 6  |-  ( -.  x  =/=  Z  <->  x  =  Z )
32ralbii 2737 . . . . 5  |-  ( A. x  e.  X  -.  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
41, 3bitr3i 251 . . . 4  |-  ( -. 
E. x  e.  X  x  =/=  Z  <->  A. x  e.  X  x  =  Z )
5 fveq2 5689 . . . . . . . . . 10  |-  ( x  =  Z  ->  ( T `  x )  =  ( T `  Z ) )
6 lnon0.1 . . . . . . . . . . 11  |-  X  =  ( BaseSet `  U )
7 eqid 2441 . . . . . . . . . . 11  |-  ( BaseSet `  W )  =  (
BaseSet `  W )
8 lnon0.6 . . . . . . . . . . 11  |-  Z  =  ( 0vec `  U
)
9 eqid 2441 . . . . . . . . . . 11  |-  ( 0vec `  W )  =  (
0vec `  W )
10 lnon0.7 . . . . . . . . . . 11  |-  L  =  ( U  LnOp  W
)
116, 7, 8, 9, 10lno0 24154 . . . . . . . . . 10  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T `  Z )  =  ( 0vec `  W
) )
125, 11sylan9eqr 2495 . . . . . . . . 9  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  x  =  Z )  ->  ( T `  x )  =  ( 0vec `  W
) )
1312ex 434 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  (
x  =  Z  -> 
( T `  x
)  =  ( 0vec `  W ) ) )
1413ralimdv 2793 . . . . . . 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 24153 . . . . . . . 8  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  T : X --> ( BaseSet `  W
) )
16 ffn 5557 . . . . . . . 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 543 . . . . . 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 5938 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) ) )
20 fvex 5699 . . . . . . . 8  |-  ( 0vec `  W )  e.  _V
2120fconst2 5932 . . . . . . 7  |-  ( T : X --> { (
0vec `  W ) } 
<->  T  =  ( X  X.  { ( 0vec `  W ) } ) )
2219, 21bitr3i 251 . . . . . 6  |-  ( ( T  Fn  X  /\  A. x  e.  X  ( T `  x )  =  ( 0vec `  W
) )  <->  T  =  ( X  X.  { (
0vec `  W ) } ) )
2318, 22syl6ib 226 . . . . 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 24185 . . . . . . 7  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
26253adant3 1008 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  O  =  ( X  X.  { ( 0vec `  W
) } ) )
2726eqeq2d 2452 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =  O  <->  T  =  ( X  X.  { (
0vec `  W ) } ) ) )
2823, 27sylibrd 234 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( A. x  e.  X  x  =  Z  ->  T  =  O ) )
294, 28syl5bi 217 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( -.  E. x  e.  X  x  =/=  Z  ->  T  =  O ) )
3029necon1ad 2676 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  ->  ( T  =/=  O  ->  E. x  e.  X  x  =/=  Z ) )
3130imp 429 1  |-  ( ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  T  e.  L )  /\  T  =/=  O )  ->  E. x  e.  X  x  =/=  Z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   A.wral 2713   E.wrex 2714   {csn 3875    X. cxp 4836    Fn wfn 5411   -->wf 5412   ` cfv 5416  (class class class)co 6089   NrmCVeccnv 23960   BaseSetcba 23962   0veccn0v 23964    LnOp clno 24138    0op c0o 24141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-1st 6575  df-2nd 6576  df-er 7099  df-map 7214  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-ltxr 9421  df-sub 9595  df-neg 9596  df-grpo 23676  df-gid 23677  df-ginv 23678  df-ablo 23767  df-vc 23922  df-nv 23968  df-va 23971  df-ba 23972  df-sm 23973  df-0v 23974  df-nmcv 23976  df-lno 24142  df-0o 24145
This theorem is referenced by: (None)
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