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Theorem cnegvex2 25763
Description: Existence of a left inverse for vector addition. (Contributed by FL, 15-Sep-2013.)
Hypotheses
Ref Expression
cnegvex2.1  |-  + w  =  (  + cv `  N )
cnegvex2.2  |-  0 w  =  ( 0 cv
`  N )
cnegvex2.3  |-  N  e.  NN
Assertion
Ref Expression
cnegvex2  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  E. x  e.  ( CC  ^m  (
1 ... N ) ) ( x + w A )  =  0 w )
Distinct variable groups:    x, A    x, N
Allowed substitution hints:    + w( x)    0 w( x)

Proof of Theorem cnegvex2
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 ax-icn 8812 . . . . . . 7  |-  _i  e.  CC
21, 1mulcli 8858 . . . . . 6  |-  ( _i  x.  _i )  e.  CC
3 cnex 8834 . . . . . . . 8  |-  CC  e.  _V
4 ovex 5899 . . . . . . . 8  |-  ( 1 ... N )  e. 
_V
53, 4elmap 6812 . . . . . . 7  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> CC )
6 ffvelrn 5679 . . . . . . 7  |-  ( ( A : ( 1 ... N ) --> CC 
/\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
75, 6sylanb 458 . . . . . 6  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
8 mulcl 8837 . . . . . 6  |-  ( ( ( _i  x.  _i )  e.  CC  /\  ( A `  i )  e.  CC )  ->  (
( _i  x.  _i )  x.  ( A `  i ) )  e.  CC )
92, 7, 8sylancr 644 . . . . 5  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( ( _i  x.  _i )  x.  ( A `  i
) )  e.  CC )
10 eqid 2296 . . . . 5  |-  ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) )
119, 10fmptd 5700 . . . 4  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  (
i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) : ( 1 ... N ) --> CC )
123, 4elmap 6812 . . . 4  |-  ( ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) )  e.  ( CC  ^m  ( 1 ... N
) )  <->  ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) ) : ( 1 ... N
) --> CC )
1311, 12sylibr 203 . . 3  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  (
i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) )  e.  ( CC  ^m  ( 1 ... N
) ) )
14 ax-1cn 8811 . . . . . . 7  |-  1  e.  CC
15 adddir 8846 . . . . . . 7  |-  ( ( ( _i  x.  _i )  e.  CC  /\  1  e.  CC  /\  ( A `
 i )  e.  CC )  ->  (
( ( _i  x.  _i )  +  1
)  x.  ( A `
 i ) )  =  ( ( ( _i  x.  _i )  x.  ( A `  i ) )  +  ( 1  x.  ( A `  i )
) ) )
162, 14, 15mp3an12 1267 . . . . . 6  |-  ( ( A `  i )  e.  CC  ->  (
( ( _i  x.  _i )  +  1
)  x.  ( A `
 i ) )  =  ( ( ( _i  x.  _i )  x.  ( A `  i ) )  +  ( 1  x.  ( A `  i )
) ) )
17 ax-i2m1 8821 . . . . . . . 8  |-  ( ( _i  x.  _i )  +  1 )  =  0
1817oveq1i 5884 . . . . . . 7  |-  ( ( ( _i  x.  _i )  +  1 )  x.  ( A `  i ) )  =  ( 0  x.  ( A `  i )
)
19 mul02 9006 . . . . . . 7  |-  ( ( A `  i )  e.  CC  ->  (
0  x.  ( A `
 i ) )  =  0 )
2018, 19syl5eq 2340 . . . . . 6  |-  ( ( A `  i )  e.  CC  ->  (
( ( _i  x.  _i )  +  1
)  x.  ( A `
 i ) )  =  0 )
21 mulid2 8852 . . . . . . 7  |-  ( ( A `  i )  e.  CC  ->  (
1  x.  ( A `
 i ) )  =  ( A `  i ) )
2221oveq2d 5890 . . . . . 6  |-  ( ( A `  i )  e.  CC  ->  (
( ( _i  x.  _i )  x.  ( A `  i )
)  +  ( 1  x.  ( A `  i ) ) )  =  ( ( ( _i  x.  _i )  x.  ( A `  i ) )  +  ( A `  i
) ) )
2316, 20, 223eqtr3rd 2337 . . . . 5  |-  ( ( A `  i )  e.  CC  ->  (
( ( _i  x.  _i )  x.  ( A `  i )
)  +  ( A `
 i ) )  =  0 )
247, 23syl 15 . . . 4  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( ( ( _i  x.  _i )  x.  ( A `  i ) )  +  ( A `  i
) )  =  0 )
2524mpteq2dva 4122 . . 3  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  (
i  e.  ( 1 ... N )  |->  ( ( ( _i  x.  _i )  x.  ( A `  i )
)  +  ( A `
 i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) )
26 nfmpt1 4125 . . . . . . 7  |-  F/_ i
( i  e.  ( 1 ... N ) 
|->  ( ( _i  x.  _i )  x.  ( A `  i )
) )
2726nfeq2 2443 . . . . . 6  |-  F/ i  x  =  ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) )
28 ovex 5899 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( A `  i ) )  e. 
_V
29 fveq1 5540 . . . . . . . . 9  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) )  -> 
( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) `
 i ) )
3010fvmpt2 5624 . . . . . . . . . . . 12  |-  ( ( i  e.  ( 1 ... N )  /\  ( ( _i  x.  _i )  x.  ( A `  i )
)  e.  _V )  ->  ( ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) ) `  i )  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) )
31 eqtr 2313 . . . . . . . . . . . . 13  |-  ( ( ( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) `
 i )  /\  ( ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) ) `  i )  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) )  ->  ( x `  i )  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) )
3231expcom 424 . . . . . . . . . . . 12  |-  ( ( ( i  e.  ( 1 ... N ) 
|->  ( ( _i  x.  _i )  x.  ( A `  i )
) ) `  i
)  =  ( ( _i  x.  _i )  x.  ( A `  i ) )  -> 
( ( x `  i )  =  ( ( i  e.  ( 1 ... N ) 
|->  ( ( _i  x.  _i )  x.  ( A `  i )
) ) `  i
)  ->  ( x `  i )  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) )
3330, 32syl 15 . . . . . . . . . . 11  |-  ( ( i  e.  ( 1 ... N )  /\  ( ( _i  x.  _i )  x.  ( A `  i )
)  e.  _V )  ->  ( ( x `  i )  =  ( ( i  e.  ( 1 ... N ) 
|->  ( ( _i  x.  _i )  x.  ( A `  i )
) ) `  i
)  ->  ( x `  i )  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) )
3433ex 423 . . . . . . . . . 10  |-  ( i  e.  ( 1 ... N )  ->  (
( ( _i  x.  _i )  x.  ( A `  i )
)  e.  _V  ->  ( ( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) `
 i )  -> 
( x `  i
)  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) ) )
3534com3l 75 . . . . . . . . 9  |-  ( ( ( _i  x.  _i )  x.  ( A `  i ) )  e. 
_V  ->  ( ( x `
 i )  =  ( ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) ) `  i )  ->  (
i  e.  ( 1 ... N )  -> 
( x `  i
)  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) ) )
3628, 29, 35mpsyl 59 . . . . . . . 8  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) )  -> 
( i  e.  ( 1 ... N )  ->  ( x `  i )  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) ) )
3736imp 418 . . . . . . 7  |-  ( ( x  =  ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) )  /\  i  e.  ( 1 ... N ) )  ->  ( x `  i )  =  ( ( _i  x.  _i )  x.  ( A `  i ) ) )
3837oveq1d 5889 . . . . . 6  |-  ( ( x  =  ( i  e.  ( 1 ... N )  |->  ( ( _i  x.  _i )  x.  ( A `  i ) ) )  /\  i  e.  ( 1 ... N ) )  ->  ( (
x `  i )  +  ( A `  i ) )  =  ( ( ( _i  x.  _i )  x.  ( A `  i
) )  +  ( A `  i ) ) )
3927, 38mpteq2da 4121 . . . . 5  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) )  -> 
( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N )  |->  ( ( ( _i  x.  _i )  x.  ( A `  i )
)  +  ( A `
 i ) ) ) )
4039eqeq1d 2304 . . . 4  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  ( ( _i  x.  _i )  x.  ( A `  i
) ) )  -> 
( ( i  e.  ( 1 ... N
)  |->  ( ( x `
 i )  +  ( A `  i
) ) )  =  ( i  e.  ( 1 ... N ) 
|->  0 )  <->  ( i  e.  ( 1 ... N
)  |->  ( ( ( _i  x.  _i )  x.  ( A `  i ) )  +  ( A `  i
) ) )  =  ( i  e.  ( 1 ... N ) 
|->  0 ) ) )
4140rspcev 2897 . . 3  |-  ( ( ( i  e.  ( 1 ... N ) 
|->  ( ( _i  x.  _i )  x.  ( A `  i )
) )  e.  ( CC  ^m  ( 1 ... N ) )  /\  ( i  e.  ( 1 ... N
)  |->  ( ( ( _i  x.  _i )  x.  ( A `  i ) )  +  ( A `  i
) ) )  =  ( i  e.  ( 1 ... N ) 
|->  0 ) )  ->  E. x  e.  ( CC  ^m  ( 1 ... N ) ) ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) )
4213, 25, 41syl2anc 642 . 2  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  E. x  e.  ( CC  ^m  (
1 ... N ) ) ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) )
43 cnegvex2.1 . . . . 5  |-  + w  =  (  + cv `  N )
44 cnegvex2.3 . . . . . . . 8  |-  N  e.  NN
45 eqid 2296 . . . . . . . . 9  |-  (  + cv `  N )  =  (  + cv `  N )
4645isaddrv 25749 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  ( CC  ^m  ( 1 ... N
) )  /\  A  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( x
(  + cv `  N
) A )  =  ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) ) )
4744, 46mp3an1 1264 . . . . . . 7  |-  ( ( x  e.  ( CC 
^m  ( 1 ... N ) )  /\  A  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( x (  + cv `  N ) A )  =  ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) ) )
4847ancoms 439 . . . . . 6  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  x  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( x (  + cv `  N ) A )  =  ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) ) )
49 oveq 5880 . . . . . . 7  |-  ( + w  =  (  + cv `  N )  ->  ( x + w A )  =  ( x (  + cv `  N ) A ) )
5049eqeq1d 2304 . . . . . 6  |-  ( + w  =  (  + cv `  N )  ->  ( ( x + w A )  =  ( i  e.  ( 1 ... N
)  |->  ( ( x `
 i )  +  ( A `  i
) ) )  <->  ( x
(  + cv `  N
) A )  =  ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) ) ) )
5148, 50syl5ibr 212 . . . . 5  |-  ( + w  =  (  + cv `  N )  ->  ( ( A  e.  ( CC  ^m  ( 1 ... N
) )  /\  x  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( x + w A )  =  ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) ) ) )
5243, 51ax-mp 8 . . . 4  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  x  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( x + w A )  =  ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) ) )
53 cnegvex2.2 . . . . . 6  |-  0 w  =  ( 0 cv
`  N )
54 eqid 2296 . . . . . . . 8  |-  ( 0 cv `  N )  =  ( 0 cv
`  N )
5554isnullcv 25755 . . . . . . 7  |-  ( N  e.  NN  ->  (
0 cv `  N
)  =  ( i  e.  ( 1 ... N )  |->  0 ) )
5644, 55ax-mp 8 . . . . . 6  |-  ( 0 cv `  N )  =  ( i  e.  ( 1 ... N
)  |->  0 )
5753, 56eqtri 2316 . . . . 5  |-  0 w  =  ( i  e.  ( 1 ... N
)  |->  0 )
5857a1i 10 . . . 4  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  x  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
0 w  =  ( i  e.  ( 1 ... N )  |->  0 ) )
5952, 58eqeq12d 2310 . . 3  |-  ( ( A  e.  ( CC 
^m  ( 1 ... N ) )  /\  x  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ( x + w A )  =  0 w  <->  ( i  e.  ( 1 ... N
)  |->  ( ( x `
 i )  +  ( A `  i
) ) )  =  ( i  e.  ( 1 ... N ) 
|->  0 ) ) )
6059rexbidva 2573 . 2  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( E. x  e.  ( CC  ^m  ( 1 ... N ) ) ( x + w A
)  =  0 w  <->  E. x  e.  ( CC 
^m  ( 1 ... N ) ) ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) ) )
6142, 60mpbird 223 1  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  ->  E. x  e.  ( CC  ^m  (
1 ... N ) ) ( x + w A )  =  0 w )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   0cc0 8753   1c1 8754   _ici 8755    + caddc 8756    x. cmul 8758   NNcn 9762   ...cfz 10798    + cvcplcv 25747   0 cvc0cv 25753
This theorem is referenced by:  cnegvex2b  25765  addcanri  25769
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-13 1698  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-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829
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 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-nel 2462  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-pw 3640  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-po 4330  df-so 4331  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-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-ltxr 8888  df-addcv 25748  df-nullcv 25754
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