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Theorem 0oval 25476
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 25475 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
54fveq1d 5868 . . 3  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
653adant3 1016 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  ( ( X  X.  { Z }
) `  A )
)
7 fvex 5876 . . . . 5  |-  ( 0vec `  W )  e.  _V
82, 7eqeltri 2551 . . . 4  |-  Z  e. 
_V
98fvconst2 6117 . . 3  |-  ( A  e.  X  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
1093ad2ant3 1019 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  (
( X  X.  { Z } ) `  A
)  =  Z )
116, 10eqtrd 2508 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec  /\  A  e.  X )  ->  ( O `  A )  =  Z )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   ` cfv 5588  (class class class)co 6285   NrmCVeccnv 25250   BaseSetcba 25252   0veccn0v 25254    0op c0o 25431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-0o 25435
This theorem is referenced by:  0lno  25478  nmoo0  25479  nmlno0lem  25481
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