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Theorem 0oval 22290
Description: Value of the zero operator. (Contributed by NM, 28-Nov-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
0oval.1  |-  X  =  ( BaseSet `  U )
0oval.6  |-  Z  =  ( 0vec `  W
)
0oval.0  |-  O  =  ( U  0op  W
)
Assertion
Ref Expression
0oval  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  Z )

Proof of Theorem 0oval
StepHypRef Expression
1 0oval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 0oval.6 . . . . 5  |-  Z  =  ( 0vec `  W
)
3 0oval.0 . . . . 5  |-  O  =  ( U  0op  W
)
41, 2, 30ofval 22289 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
54fveq1d 5731 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
653adant3 978 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
7 fvex 5743 . . . . 5  |-  ( 0vec `  W )  e.  _V
82, 7eqeltri 2507 . . . 4  |-  Z  e. 
_V
98fvconst2 5948 . . 3  |-  ( A  e.  X  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
1093ad2ant3 981 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
116, 10eqtrd 2469 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   _Vcvv 2957   {csn 3815    X. cxp 4877   ` cfv 5455  (class class class)co 6082   NrmCVeccnv 22064   BaseSetcba 22066   0veccn0v 22068    0op c0o 22245
This theorem is referenced by:  0lno  22292  nmoo0  22293  nmlno0lem  22295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-0o 22249
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