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Theorem 0ofval 25375
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 5864 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 0oval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2526 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54xpeq1d 5022 . . 3  |-  ( u  =  U  ->  (
( BaseSet `  u )  X.  { ( 0vec `  w
) } )  =  ( X  X.  {
( 0vec `  w ) } ) )
6 fveq2 5864 . . . . . 6  |-  ( w  =  W  ->  ( 0vec `  w )  =  ( 0vec `  W
) )
7 0oval.6 . . . . . 6  |-  Z  =  ( 0vec `  W
)
86, 7syl6eqr 2526 . . . . 5  |-  ( w  =  W  ->  ( 0vec `  w )  =  Z )
98sneqd 4039 . . . 4  |-  ( w  =  W  ->  { (
0vec `  w ) }  =  { Z } )
109xpeq2d 5023 . . 3  |-  ( w  =  W  ->  ( X  X.  { ( 0vec `  w ) } )  =  ( X  X.  { Z } ) )
11 df-0o 25335 . . 3  |-  0op  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  ( ( BaseSet `  u )  X.  {
( 0vec `  w ) } ) )
12 fvex 5874 . . . . 5  |-  ( BaseSet `  U )  e.  _V
133, 12eqeltri 2551 . . . 4  |-  X  e. 
_V
14 snex 4688 . . . 4  |-  { Z }  e.  _V
1513, 14xpex 6711 . . 3  |-  ( X  X.  { Z }
)  e.  _V
165, 10, 11, 15ovmpt2 6420 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  0op  W )  =  ( X  X.  { Z } ) )
171, 16syl5eq 2520 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   {csn 4027    X. cxp 4997   ` cfv 5586  (class class class)co 6282   NrmCVeccnv 25150   BaseSetcba 25152   0veccn0v 25154    0op c0o 25331
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 6574
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-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-0o 25335
This theorem is referenced by:  0oval  25376  0oo  25377  lnon0  25386  blocni  25393  hh0oi  26495
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