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Theorem 0oval 22138
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 22137 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
54fveq1d 5671 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
653adant3 977 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
7 fvex 5683 . . . . 5  |-  ( 0vec `  W )  e.  _V
82, 7eqeltri 2458 . . . 4  |-  Z  e. 
_V
98fvconst2 5887 . . 3  |-  ( A  e.  X  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
1093ad2ant3 980 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
116, 10eqtrd 2420 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2900   {csn 3758    X. cxp 4817   ` cfv 5395  (class class class)co 6021   NrmCVeccnv 21912   BaseSetcba 21914   0veccn0v 21916    0op c0o 22093
This theorem is referenced by:  0lno  22140  nmoo0  22141  nmlno0lem  22143
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-0o 22097
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