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Theorem 0ofval 22319
Description: The zero operator between two normed complex vector spaces. (Contributed by NM, 28-Nov-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (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
0ofval  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )

Proof of Theorem 0ofval
Dummy variables  w  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0oval.0 . 2  |-  O  =  ( U  0op  W
)
2 fveq2 5757 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 0oval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2492 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54xpeq1d 4930 . . 3  |-  ( u  =  U  ->  (
( BaseSet `  u )  X.  { ( 0vec `  w
) } )  =  ( X  X.  {
( 0vec `  w ) } ) )
6 fveq2 5757 . . . . . 6  |-  ( w  =  W  ->  ( 0vec `  w )  =  ( 0vec `  W
) )
7 0oval.6 . . . . . 6  |-  Z  =  ( 0vec `  W
)
86, 7syl6eqr 2492 . . . . 5  |-  ( w  =  W  ->  ( 0vec `  w )  =  Z )
98sneqd 3851 . . . 4  |-  ( w  =  W  ->  { (
0vec `  w ) }  =  { Z } )
109xpeq2d 4931 . . 3  |-  ( w  =  W  ->  ( X  X.  { ( 0vec `  w ) } )  =  ( X  X.  { Z } ) )
11 df-0o 22279 . . 3  |-  0op  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  ( ( BaseSet `  u )  X.  {
( 0vec `  w ) } ) )
12 fvex 5771 . . . . 5  |-  ( BaseSet `  U )  e.  _V
133, 12eqeltri 2512 . . . 4  |-  X  e. 
_V
14 snex 4434 . . . 4  |-  { Z }  e.  _V
1513, 14xpex 5019 . . 3  |-  ( X  X.  { Z }
)  e.  _V
165, 10, 11, 15ovmpt2 6238 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  0op  W )  =  ( X  X.  { Z } ) )
171, 16syl5eq 2486 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727   _Vcvv 2962   {csn 3838    X. cxp 4905   ` cfv 5483  (class class class)co 6110   NrmCVeccnv 22094   BaseSetcba 22096   0veccn0v 22098    0op c0o 22275
This theorem is referenced by:  0oval  22320  0oo  22321  lnon0  22330  blocni  22337  hh0oi  23437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
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 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-0o 22279
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