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Theorem rnegvex2 25764
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 5679 . . . . . 6  |-  ( ( A : ( 1 ... N ) --> RR 
/\  i  e.  ( 1 ... N ) )  ->  ( A `  i )  e.  RR )
21renegcld 9226 . . . . 5  |-  ( ( A : ( 1 ... N ) --> RR 
/\  i  e.  ( 1 ... N ) )  ->  -u ( A `
 i )  e.  RR )
3 eqid 2296 . . . . 5  |-  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)  =  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)
42, 3fmptd 5700 . . . 4  |-  ( A : ( 1 ... N ) --> RR  ->  ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) ) : ( 1 ... N ) --> RR )
5 reex 8844 . . . . 5  |-  RR  e.  _V
6 ovex 5899 . . . . 5  |-  ( 1 ... N )  e. 
_V
75, 6elmap 6812 . . . 4  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  <->  A :
( 1 ... N
) --> RR )
85, 6elmap 6812 . . . 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 8877 . . . . . . . 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 9160 . . . . . 6  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  -u ( A `  i )  e.  CC )
1312, 11addcomd 9030 . . . . 5  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( -u ( A `  i )  +  ( A `  i ) )  =  ( ( A `  i )  +  -u ( A `  i ) ) )
1411negidd 9163 . . . . 5  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( ( A `
 i )  + 
-u ( A `  i ) )  =  0 )
1513, 14eqtrd 2328 . . . 4  |-  ( ( A  e.  ( RR 
^m  ( 1 ... N ) )  /\  i  e.  ( 1 ... N ) )  ->  ( -u ( A `  i )  +  ( A `  i ) )  =  0 )
1615mpteq2dva 4122 . . 3  |-  ( A  e.  ( RR  ^m  ( 1 ... N
) )  ->  (
i  e.  ( 1 ... N )  |->  (
-u ( A `  i )  +  ( A `  i ) ) )  =  ( i  e.  ( 1 ... N )  |->  0 ) )
17 nfmpt1 4125 . . . . . . 7  |-  F/_ i
( i  e.  ( 1 ... N ) 
|->  -u ( A `  i ) )
1817nfeq2 2443 . . . . . 6  |-  F/ i  x  =  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)
19 negex 9066 . . . . . . . . 9  |-  -u ( A `  i )  e.  _V
20 fveq1 5540 . . . . . . . . 9  |-  ( x  =  ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) )  -> 
( x `  i
)  =  ( ( i  e.  ( 1 ... N )  |->  -u ( A `  i ) ) `  i ) )
213fvmpt2 5624 . . . . . . . . . . . 12  |-  ( ( i  e.  ( 1 ... N )  /\  -u ( A `  i
)  e.  _V )  ->  ( ( i  e.  ( 1 ... N
)  |->  -u ( A `  i ) ) `  i )  =  -u ( A `  i ) )
22 eqtr 2313 . . . . . . . . . . . . 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 5889 . . . . . 6  |-  ( ( x  =  ( i  e.  ( 1 ... N )  |->  -u ( A `  i )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  +  ( A `
 i ) )  =  ( -u ( A `  i )  +  ( A `  i ) ) )
3018, 29mpteq2da 4121 . . . . 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 2304 . . . 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 2897 . . 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 8810 . . . . . . . . . . 11  |-  RR  C_  CC
38 fss 5413 . . . . . . . . . . 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 6812 . . . . . . . . . 10  |-  ( x  e.  ( RR  ^m  ( 1 ... N
) )  <->  x :
( 1 ... N
) --> RR )
41 cnex 8834 . . . . . . . . . . 11  |-  CC  e.  _V
4241, 6elmap 6812 . . . . . . . . . 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 5413 . . . . . . . . . . 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 6812 . . . . . . . . . 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 2296 . . . . . . . . . 10  |-  (  + cv `  N )  =  (  + cv `  N )
4948isaddrv 25749 . . . . . . . . 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 5880 . . . . . . 7  |-  ( + w  =  (  + cv `  N )  ->  ( x + w A )  =  ( x (  + cv `  N ) A ) )
5453eqeq1d 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 ) ) ) ) )
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 25755 . . . . 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 2310 . . 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 2573 . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165    e. cmpt 4093   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756   -ucneg 9054   NNcn 9762   ...cfz 10798    + cvcplcv 25747   0 cvc0cv 25753
This theorem is referenced by:  rnegvex2b  25766
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-riota 6320  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-sub 9055  df-neg 9056  df-addcv 25748  df-nullcv 25754
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