MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0oval Structured version   Unicode version

Theorem 0oval 22279
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 22278 . . . 4  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
54fveq1d 5722 . . 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 5734 . . . . 5  |-  ( 0vec `  W )  e.  _V
82, 7eqeltri 2505 . . . 4  |-  Z  e. 
_V
98fvconst2 5939 . . 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 2467 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 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    X. cxp 4868   ` cfv 5446  (class class class)co 6073   NrmCVeccnv 22053   BaseSetcba 22055   0veccn0v 22057    0op c0o 22234
This theorem is referenced by:  0lno  22281  nmoo0  22282  nmlno0lem  22284
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-0o 22238
  Copyright terms: Public domain W3C validator