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Theorem addidv2 25657
Description: The null vector is a left identity for vector addition. (Contributed by FL, 15-Sep-2013.)
Hypotheses
Ref Expression
addidv2.1  |-  0 w  =  ( 0 cv
`  N )
addidv2.2  |-  + w  =  (  + cv `  N )
Assertion
Ref Expression
addidv2  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( 0 w + w A )  =  A )

Proof of Theorem addidv2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 addidv2.1 . . . . . . 7  |-  0 w  =  ( 0 cv
`  N )
21valvze 25654 . . . . . 6  |-  ( ( N  e.  NN  /\  x  e.  ( 1 ... N ) )  ->  ( 0 w `  x )  =  0 )
32adantlr 695 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ( CC 
^m  ( 1 ... N ) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( 0 w `  x )  =  0 )
43oveq1d 5873 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ( CC 
^m  ( 1 ... N ) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( (
0 w `  x
)  +  ( A `
 x ) )  =  ( 0  +  ( A `  x
) ) )
5 cnex 8818 . . . . . . . 8  |-  CC  e.  _V
6 ovex 5883 . . . . . . . 8  |-  ( 1 ... N )  e. 
_V
75, 6elmap 6796 . . . . . . 7  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> CC )
8 ffvelrn 5663 . . . . . . . . 9  |-  ( ( A : ( 1 ... N ) --> CC 
/\  x  e.  ( 1 ... N ) )  ->  ( A `  x )  e.  CC )
98ex 423 . . . . . . . 8  |-  ( A : ( 1 ... N ) --> CC  ->  ( x  e.  ( 1 ... N )  -> 
( A `  x
)  e.  CC ) )
109a1i 10 . . . . . . 7  |-  ( N  e.  NN  ->  ( A : ( 1 ... N ) --> CC  ->  ( x  e.  ( 1 ... N )  -> 
( A `  x
)  e.  CC ) ) )
117, 10syl5bi 208 . . . . . 6  |-  ( N  e.  NN  ->  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  (
x  e.  ( 1 ... N )  -> 
( A `  x
)  e.  CC ) ) )
1211imp31 421 . . . . 5  |-  ( ( ( N  e.  NN  /\  A  e.  ( CC 
^m  ( 1 ... N ) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( A `  x )  e.  CC )
1312addid2d 9013 . . . 4  |-  ( ( ( N  e.  NN  /\  A  e.  ( CC 
^m  ( 1 ... N ) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( 0  +  ( A `  x ) )  =  ( A `  x
) )
144, 13eqtrd 2315 . . 3  |-  ( ( ( N  e.  NN  /\  A  e.  ( CC 
^m  ( 1 ... N ) ) )  /\  x  e.  ( 1 ... N ) )  ->  ( (
0 w `  x
)  +  ( A `
 x ) )  =  ( A `  x ) )
1514mpteq2dva 4106 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( x  e.  ( 1 ... N ) 
|->  ( ( 0 w `  x )  +  ( A `  x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( A `  x ) ) )
16 simpl 443 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  N  e.  NN )
171zernpl 25653 . . . 4  |-  ( N  e.  NN  ->  0 w  e.  ( CC  ^m  ( 1 ... N
) ) )
1817adantr 451 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
0 w  e.  ( CC  ^m  ( 1 ... N ) ) )
19 simpr 447 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  A  e.  ( CC  ^m  ( 1 ... N
) ) )
20 addidv2.2 . . . 4  |-  + w  =  (  + cv `  N )
2120isaddrv 25646 . . 3  |-  ( ( N  e.  NN  /\  0 w  e.  ( CC  ^m  ( 1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( 0 w + w A )  =  ( x  e.  ( 1 ... N )  |->  ( ( 0 w `  x )  +  ( A `  x ) ) ) )
2216, 18, 19, 21syl3anc 1182 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( 0 w + w A )  =  ( x  e.  ( 1 ... N )  |->  ( ( 0 w `  x )  +  ( A `  x ) ) ) )
2319, 7sylib 188 . . 3  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  A : ( 1 ... N ) --> CC )
2423feqmptd 5575 . 2  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  A  =  ( x  e.  ( 1 ... N
)  |->  ( A `  x ) ) )
2515, 22, 243eqtr4d 2325 1  |-  ( ( N  e.  NN  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( 0 w + w A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740   NNcn 9746   ...cfz 10782    + cvcplcv 25644   0 cvc0cv 25650
This theorem is referenced by:  addidrv2  25658  vecaddonto  25659  addcanri  25666  addcanrg  25667  negveud  25668  negveudr  25669  tcnvec  25690
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-13 1686  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-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-nel 2449  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-pw 3627  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-po 4314  df-so 4315  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872  df-addcv 25645  df-nullcv 25651
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