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Theorem 0ofval 24192
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 5696 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 0oval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2493 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54xpeq1d 4868 . . 3  |-  ( u  =  U  ->  (
( BaseSet `  u )  X.  { ( 0vec `  w
) } )  =  ( X  X.  {
( 0vec `  w ) } ) )
6 fveq2 5696 . . . . . 6  |-  ( w  =  W  ->  ( 0vec `  w )  =  ( 0vec `  W
) )
7 0oval.6 . . . . . 6  |-  Z  =  ( 0vec `  W
)
86, 7syl6eqr 2493 . . . . 5  |-  ( w  =  W  ->  ( 0vec `  w )  =  Z )
98sneqd 3894 . . . 4  |-  ( w  =  W  ->  { (
0vec `  w ) }  =  { Z } )
109xpeq2d 4869 . . 3  |-  ( w  =  W  ->  ( X  X.  { ( 0vec `  w ) } )  =  ( X  X.  { Z } ) )
11 df-0o 24152 . . 3  |-  0op  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  ( ( BaseSet `  u )  X.  {
( 0vec `  w ) } ) )
12 fvex 5706 . . . . 5  |-  ( BaseSet `  U )  e.  _V
133, 12eqeltri 2513 . . . 4  |-  X  e. 
_V
14 snex 4538 . . . 4  |-  { Z }  e.  _V
1513, 14xpex 6513 . . 3  |-  ( X  X.  { Z }
)  e.  _V
165, 10, 11, 15ovmpt2 6231 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  0op  W )  =  ( X  X.  { Z } ) )
171, 16syl5eq 2487 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   _Vcvv 2977   {csn 3882    X. cxp 4843   ` cfv 5423  (class class class)co 6096   NrmCVeccnv 23967   BaseSetcba 23969   0veccn0v 23971    0op c0o 24148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5386  df-fun 5425  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-0o 24152
This theorem is referenced by:  0oval  24193  0oo  24194  lnon0  24203  blocni  24210  hh0oi  25312
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