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Theorem 0ofval 26273
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 5881 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 0oval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2488 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54xpeq1d 4877 . . 3  |-  ( u  =  U  ->  (
( BaseSet `  u )  X.  { ( 0vec `  w
) } )  =  ( X  X.  {
( 0vec `  w ) } ) )
6 fveq2 5881 . . . . . 6  |-  ( w  =  W  ->  ( 0vec `  w )  =  ( 0vec `  W
) )
7 0oval.6 . . . . . 6  |-  Z  =  ( 0vec `  W
)
86, 7syl6eqr 2488 . . . . 5  |-  ( w  =  W  ->  ( 0vec `  w )  =  Z )
98sneqd 4014 . . . 4  |-  ( w  =  W  ->  { (
0vec `  w ) }  =  { Z } )
109xpeq2d 4878 . . 3  |-  ( w  =  W  ->  ( X  X.  { ( 0vec `  w ) } )  =  ( X  X.  { Z } ) )
11 df-0o 26233 . . 3  |-  0op  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  ( ( BaseSet `  u )  X.  {
( 0vec `  w ) } ) )
12 fvex 5891 . . . . 5  |-  ( BaseSet `  U )  e.  _V
133, 12eqeltri 2513 . . . 4  |-  X  e. 
_V
14 snex 4663 . . . 4  |-  { Z }  e.  _V
1513, 14xpex 6609 . . 3  |-  ( X  X.  { Z }
)  e.  _V
165, 10, 11, 15ovmpt2 6446 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  0op  W )  =  ( X  X.  { Z } ) )
171, 16syl5eq 2482 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   _Vcvv 3087   {csn 4002    X. cxp 4852   ` cfv 5601  (class class class)co 6305   NrmCVeccnv 26048   BaseSetcba 26050   0veccn0v 26052    0op c0o 26229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-0o 26233
This theorem is referenced by:  0oval  26274  0oo  26275  lnon0  26284  blocni  26291  hh0oi  27391
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