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Theorem rnegvex2 25661
Description: Existence of a left inverse for vector addition. (Contributed by FL, 29-May-2014.)
Hypotheses
Ref Expression
cnegvex2.1  |-  + w  =  (  + cv `  N )
cnegvex2.2  |-  0 w  =  ( 0 cv
`  N )
cnegvex2.3  |-  N  e.  NN
Assertion
Ref Expression
rnegvex2  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  E. x  e.  ( RR  ^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 rnegvex2
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 ffvelrn 5663 . . . . . 6  |-  ( ( A : ( 1 ... N ) --> RR 
/\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
21renegcld 9210 . . . . 5  |-  ( ( A : ( 1 ... N ) --> RR 
/\  i  e.  ( 1 ... N ) )  ->  -u ( A `
 i )  e.  RR )
3 eqid 2283 . . . . 5  |-  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)  =  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)
42, 3fmptd 5684 . . . 4  |-  ( A : ( 1 ... N ) --> RR  ->  ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) ) : ( 1 ... N ) --> RR )
5 reex 8828 . . . . 5  |-  RR  e.  _V
6 ovex 5883 . . . . 5  |-  ( 1 ... N )  e. 
_V
75, 6elmap 6796 . . . 4  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> RR )
85, 6elmap 6796 . . . 4  |-  ( ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) )  e.  ( RR 
^m  ( 1 ... N ) )  <->  ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) ) : ( 1 ... N
) --> RR )
94, 7, 83imtr4i 257 . . 3  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  (
i  e.  ( 1 ... N )  |->  -u ( A `  i ) )  e.  ( RR 
^m  ( 1 ... N ) ) )
101recnd 8861 . . . . . . . 8  |-  ( ( A : ( 1 ... N ) --> RR 
/\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
117, 10sylanb 458 . . . . . . 7  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  CC )
1211negcld 9144 . . . . . 6  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  -u ( A `  i )  e.  CC )
1312, 11addcomd 9014 . . . . 5  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( -u ( A `  i )  +  ( A `  i ) )  =  ( ( A `  i )  +  -u ( A `  i ) ) )
1411negidd 9147 . . . . 5  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( ( A `
 i )  + 
-u ( A `  i ) )  =  0 )
1513, 14eqtrd 2315 . . . 4  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( -u ( A `  i )  +  ( A `  i ) )  =  0 )
1615mpteq2dva 4106 . . 3  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  (
i  e.  ( 1 ... N )  |->  (
-u ( A `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) )
17 nfmpt1 4109 . . . . . . 7  |-  F/_ i
( i  e.  ( 1 ... N ) 
|->  -u ( A `  i ) )
1817nfeq2 2430 . . . . . 6  |-  F/ i  x  =  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)
19 negex 9050 . . . . . . . . 9  |-  -u ( A `  i )  e.  _V
20 fveq1 5524 . . . . . . . . 9  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) )  -> 
( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) ) `  i ) )
213fvmpt2 5608 . . . . . . . . . . . 12  |-  ( ( i  e.  ( 1 ... N )  /\  -u ( A `  i
)  e.  _V )  ->  ( ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) ) `  i )  =  -u ( A `  i ) )
22 eqtr 2300 . . . . . . . . . . . . 13  |-  ( ( ( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) ) `  i )  /\  ( ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
) `  i )  =  -u ( A `  i ) )  -> 
( x `  i
)  =  -u ( A `  i )
)
2322expcom 424 . . . . . . . . . . . 12  |-  ( ( ( i  e.  ( 1 ... N ) 
|->  -u ( A `  i ) ) `  i )  =  -u ( A `  i )  ->  ( ( x `
 i )  =  ( ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) ) `  i )  ->  (
x `  i )  =  -u ( A `  i ) ) )
2421, 23syl 15 . . . . . . . . . . 11  |-  ( ( i  e.  ( 1 ... N )  /\  -u ( A `  i
)  e.  _V )  ->  ( ( x `  i )  =  ( ( i  e.  ( 1 ... N ) 
|->  -u ( A `  i ) ) `  i )  ->  (
x `  i )  =  -u ( A `  i ) ) )
2524ex 423 . . . . . . . . . 10  |-  ( i  e.  ( 1 ... N )  ->  ( -u ( A `  i
)  e.  _V  ->  ( ( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) ) `  i )  ->  ( x `  i )  =  -u ( A `  i ) ) ) )
2625com3l 75 . . . . . . . . 9  |-  ( -u ( A `  i )  e.  _V  ->  (
( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) ) `  i )  ->  ( i  e.  ( 1 ... N
)  ->  ( x `  i )  =  -u ( A `  i ) ) ) )
2719, 20, 26mpsyl 59 . . . . . . . 8  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) )  -> 
( i  e.  ( 1 ... N )  ->  ( x `  i )  =  -u ( A `  i ) ) )
2827imp 418 . . . . . . 7  |-  ( ( x  =  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)  /\  i  e.  ( 1 ... N
) )  ->  (
x `  i )  =  -u ( A `  i ) )
2928oveq1d 5873 . . . . . 6  |-  ( ( x  =  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  +  ( A `
 i ) )  =  ( -u ( A `  i )  +  ( A `  i ) ) )
3018, 29mpteq2da 4105 . . . . 5  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) )  -> 
( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N )  |->  (
-u ( A `  i )  +  ( A `  i ) ) ) )
3130eqeq1d 2291 . . . 4  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) )  -> 
( ( i  e.  ( 1 ... N
)  |->  ( ( x `
 i )  +  ( A `  i
) ) )  =  ( i  e.  ( 1 ... N ) 
|->  0 )  <->  ( i  e.  ( 1 ... N
)  |->  ( -u ( A `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N
)  |->  0 ) ) )
3231rspcev 2884 . . 3  |-  ( ( ( i  e.  ( 1 ... N ) 
|->  -u ( A `  i ) )  e.  ( RR  ^m  (
1 ... N ) )  /\  ( i  e.  ( 1 ... N
)  |->  ( -u ( A `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N
)  |->  0 ) )  ->  E. x  e.  ( RR  ^m  ( 1 ... N ) ) ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) )
339, 16, 32syl2anc 642 . 2  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  E. x  e.  ( RR  ^m  (
1 ... N ) ) ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) )
34 cnegvex2.1 . . . . 5  |-  + w  =  (  + cv `  N )
35 cnegvex2.3 . . . . . . . 8  |-  N  e.  NN
36 id 19 . . . . . . . . 9  |-  ( N  e.  NN  ->  N  e.  NN )
37 ax-resscn 8794 . . . . . . . . . . 11  |-  RR  C_  CC
38 fss 5397 . . . . . . . . . . 11  |-  ( ( x : ( 1 ... N ) --> RR 
/\  RR  C_  CC )  ->  x : ( 1 ... N ) --> CC )
3937, 38mpan2 652 . . . . . . . . . 10  |-  ( x : ( 1 ... N ) --> RR  ->  x : ( 1 ... N ) --> CC )
405, 6elmap 6796 . . . . . . . . . 10  |-  ( x  e.  ( RR  ^m  ( 1 ... N
) )  <->  x :
( 1 ... N
) --> RR )
41 cnex 8818 . . . . . . . . . . 11  |-  CC  e.  _V
4241, 6elmap 6796 . . . . . . . . . 10  |-  ( x  e.  ( CC  ^m  ( 1 ... N
) )  <->  x :
( 1 ... N
) --> CC )
4339, 40, 423imtr4i 257 . . . . . . . . 9  |-  ( x  e.  ( RR  ^m  ( 1 ... N
) )  ->  x  e.  ( CC  ^m  (
1 ... N ) ) )
44 fss 5397 . . . . . . . . . . 11  |-  ( ( A : ( 1 ... N ) --> RR 
/\  RR  C_  CC )  ->  A : ( 1 ... N ) --> CC )
4537, 44mpan2 652 . . . . . . . . . 10  |-  ( A : ( 1 ... N ) --> RR  ->  A : ( 1 ... N ) --> CC )
4641, 6elmap 6796 . . . . . . . . . 10  |-  ( A  e.  ( CC  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> CC )
4745, 7, 463imtr4i 257 . . . . . . . . 9  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  A  e.  ( CC  ^m  (
1 ... N ) ) )
48 eqid 2283 . . . . . . . . . 10  |-  (  + cv `  N )  =  (  + cv `  N )
4948isaddrv 25646 . . . . . . . . 9  |-  ( ( 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 ) ) ) )
5036, 43, 47, 49syl3an 1224 . . . . . . . 8  |-  ( ( N  e.  NN  /\  x  e.  ( RR  ^m  ( 1 ... N
) )  /\  A  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x
(  + cv `  N
) A )  =  ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) ) )
5135, 50mp3an1 1264 . . . . . . 7  |-  ( ( x  e.  ( RR 
^m  ( 1 ... N ) )  /\  A  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
( x (  + cv `  N ) A )  =  ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) ) )
5251ancoms 439 . . . . . 6  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  x  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
( x (  + cv `  N ) A )  =  ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) ) )
53 oveq 5864 . . . . . . 7  |-  ( + w  =  (  + cv `  N )  ->  ( x + w A )  =  ( x (  + cv `  N ) A ) )
5453eqeq1d 2291 . . . . . 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 ) ) ) ) )
5552, 54syl5ibr 212 . . . . 5  |-  ( + w  =  (  + cv `  N )  ->  ( ( A  e.  ( RR  ^m  ( 1 ... N
) )  /\  x  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x + w A )  =  ( i  e.  ( 1 ... N ) 
|->  ( ( x `  i )  +  ( A `  i ) ) ) ) )
5634, 55ax-mp 8 . . . 4  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  x  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
( x + w A )  =  ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) ) )
57 cnegvex2.2 . . . . . 6  |-  0 w  =  ( 0 cv
`  N )
5857isnullcv 25652 . . . . 5  |-  ( N  e.  NN  ->  0 w  =  ( i  e.  ( 1 ... N
)  |->  0 ) )
5935, 58mp1i 11 . . . 4  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  x  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
0 w  =  ( i  e.  ( 1 ... N )  |->  0 ) )
6056, 59eqeq12d 2297 . . 3  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  x  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
( ( x + w A )  =  0 w  <->  ( i  e.  ( 1 ... N
)  |->  ( ( x `
 i )  +  ( A `  i
) ) )  =  ( i  e.  ( 1 ... N ) 
|->  0 ) ) )
6160rexbidva 2560 . 2  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  ( E. x  e.  ( RR  ^m  ( 1 ... N ) ) ( x + w A
)  =  0 w  <->  E. x  e.  ( RR 
^m  ( 1 ... N ) ) ( i  e.  ( 1 ... N )  |->  ( ( x `  i
)  +  ( A `
 i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) ) )
6233, 61mpbird 223 1  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  E. x  e.  ( RR  ^m  (
1 ... N ) ) ( x + w A )  =  0 w )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740   -ucneg 9038   NNcn 9746   ...cfz 10782    + cvcplcv 25644   0 cvc0cv 25650
This theorem is referenced by:  rnegvex2b  25663
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-riota 6304  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-sub 9039  df-neg 9040  df-addcv 25645  df-nullcv 25651
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