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Theorem isnullcv 25652
Description: The null vector in a space of dimension  n. (Contributed by FL, 15-Sep-2013.)
Hypothesis
Ref Expression
isnullcv.1  |-  0 w  =  ( 0 cv
`  N )
Assertion
Ref Expression
isnullcv  |-  ( N  e.  NN  ->  0 w  =  ( x  e.  ( 1 ... N
)  |->  0 ) )
Distinct variable group:    x, N
Allowed substitution hint:    0 w( x)

Proof of Theorem isnullcv
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 ovex 5883 . . 3  |-  ( 1 ... N )  e. 
_V
21mptex 5746 . 2  |-  ( x  e.  ( 1 ... N )  |->  0 )  e.  _V
3 isnullcv.1 . . 3  |-  0 w  =  ( 0 cv
`  N )
4 oveq2 5866 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
5 eqidd 2284 . . . . 5  |-  ( n  =  N  ->  0  =  0 )
64, 5mpteq12dv 4098 . . . 4  |-  ( n  =  N  ->  (
x  e.  ( 1 ... n )  |->  0 )  =  ( x  e.  ( 1 ... N )  |->  0 ) )
7 df-nullcv 25651 . . . 4  |-  0 cv  =  ( n  e.  NN  |->  ( x  e.  ( 1 ... n
)  |->  0 ) )
86, 7fvmptg 5600 . . 3  |-  ( ( N  e.  NN  /\  ( x  e.  (
1 ... N )  |->  0 )  e.  _V )  ->  ( 0 cv `  N
)  =  ( x  e.  ( 1 ... N )  |->  0 ) )
93, 8syl5eq 2327 . 2  |-  ( ( N  e.  NN  /\  ( x  e.  (
1 ... N )  |->  0 )  e.  _V )  ->  0 w  =  ( x  e.  ( 1 ... N )  |->  0 ) )
102, 9mpan2 652 1  |-  ( N  e.  NN  ->  0 w  =  ( x  e.  ( 1 ... N
)  |->  0 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   0cc0 8737   1c1 8738   NNcn 9746   ...cfz 10782   0 cvc0cv 25650
This theorem is referenced by:  zernpl  25653  cnegvex2  25660  rnegvex2  25661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-nullcv 25651
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