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Theorem 0ofval 22278
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 5720 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 0oval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2485 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
54xpeq1d 4893 . . 3  |-  ( u  =  U  ->  (
( BaseSet `  u )  X.  { ( 0vec `  w
) } )  =  ( X  X.  {
( 0vec `  w ) } ) )
6 fveq2 5720 . . . . . 6  |-  ( w  =  W  ->  ( 0vec `  w )  =  ( 0vec `  W
) )
7 0oval.6 . . . . . 6  |-  Z  =  ( 0vec `  W
)
86, 7syl6eqr 2485 . . . . 5  |-  ( w  =  W  ->  ( 0vec `  w )  =  Z )
98sneqd 3819 . . . 4  |-  ( w  =  W  ->  { (
0vec `  w ) }  =  { Z } )
109xpeq2d 4894 . . 3  |-  ( w  =  W  ->  ( X  X.  { ( 0vec `  w ) } )  =  ( X  X.  { Z } ) )
11 df-0o 22238 . . 3  |-  0op  =  ( u  e.  NrmCVec ,  w  e.  NrmCVec  |->  ( ( BaseSet `  u )  X.  {
( 0vec `  w ) } ) )
12 fvex 5734 . . . . 5  |-  ( BaseSet `  U )  e.  _V
133, 12eqeltri 2505 . . . 4  |-  X  e. 
_V
14 snex 4397 . . . 4  |-  { Z }  e.  _V
1513, 14xpex 4982 . . 3  |-  ( X  X.  { Z }
)  e.  _V
165, 10, 11, 15ovmpt2 6201 . 2  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  ( U  0op  W )  =  ( X  X.  { Z } ) )
171, 16syl5eq 2479 1  |-  ( ( U  e.  NrmCVec  /\  W  e.  NrmCVec )  ->  O  =  ( X  X.  { Z } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = 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:  0oval  22279  0oo  22280  lnon0  22289  blocni  22296  hh0oi  23396
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-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-0o 22238
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