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Theorem 0oval 21366
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 21365 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
54fveq1d 5527 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
653adant3 975 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
7 fvex 5539 . . . . 5  |-  ( 0vec `  W )  e.  _V
82, 7eqeltri 2353 . . . 4  |-  Z  e. 
_V
98fvconst2 5729 . . 3  |-  ( A  e.  X  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
1093ad2ant3 978 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
116, 10eqtrd 2315 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  Z )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640    X. cxp 4687   ` cfv 5255  (class class class)co 5858   NrmCVeccnv 21140   BaseSetcba 21142   0veccn0v 21144    0op c0o 21321
This theorem is referenced by:  0lno  21368  nmoo0  21369  nmlno0lem  21371
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-0o 21325
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