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Theorem isnullcv 25755
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 5899 . . 3  |-  ( 1 ... N )  e. 
_V
21mptex 5762 . 2  |-  ( x  e.  ( 1 ... N )  |->  0 )  e.  _V
3 isnullcv.1 . . 3  |-  0 w  =  ( 0 cv
`  N )
4 oveq2 5882 . . . . 5  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
5 eqidd 2297 . . . . 5  |-  ( n  =  N  ->  0  =  0 )
64, 5mpteq12dv 4114 . . . 4  |-  ( n  =  N  ->  (
x  e.  ( 1 ... n )  |->  0 )  =  ( x  e.  ( 1 ... N )  |->  0 ) )
7 df-nullcv 25754 . . . 4  |-  0 cv  =  ( n  e.  NN  |->  ( x  e.  ( 1 ... n
)  |->  0 ) )
86, 7fvmptg 5616 . . 3  |-  ( ( N  e.  NN  /\  ( x  e.  (
1 ... N )  |->  0 )  e.  _V )  ->  ( 0 cv `  N
)  =  ( x  e.  ( 1 ... N )  |->  0 ) )
93, 8syl5eq 2340 . 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 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   0cc0 8753   1c1 8754   NNcn 9762   ...cfz 10798   0 cvc0cv 25753
This theorem is referenced by:  zernpl  25756  cnegvex2  25763  rnegvex2  25764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-nullcv 25754
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