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Theorem claddrvr 25648
Description: Closure of addition of real vectors. (Contributed by FL, 29-May-2014.)
Hypothesis
Ref Expression
isaddrv.1  |-  + w  =  (  + cv `  N )
Assertion
Ref Expression
claddrvr  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( U + w V )  e.  ( RR  ^m  (
1 ... N ) ) )

Proof of Theorem claddrvr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  N  e.  NN )
2 ax-resscn 8794 . . . . . 6  |-  RR  C_  CC
3 fss 5397 . . . . . 6  |-  ( ( U : ( 1 ... N ) --> RR 
/\  RR  C_  CC )  ->  U : ( 1 ... N ) --> CC )
42, 3mpan2 652 . . . . 5  |-  ( U : ( 1 ... N ) --> RR  ->  U : ( 1 ... N ) --> CC )
5 reex 8828 . . . . . 6  |-  RR  e.  _V
6 ovex 5883 . . . . . 6  |-  ( 1 ... N )  e. 
_V
75, 6elmap 6796 . . . . 5  |-  ( U  e.  ( RR  ^m  ( 1 ... N
) )  <->  U :
( 1 ... N
) --> RR )
8 cnex 8818 . . . . . 6  |-  CC  e.  _V
98, 6elmap 6796 . . . . 5  |-  ( U  e.  ( CC  ^m  ( 1 ... N
) )  <->  U :
( 1 ... N
) --> CC )
104, 7, 93imtr4i 257 . . . 4  |-  ( U  e.  ( RR  ^m  ( 1 ... N
) )  ->  U  e.  ( CC  ^m  (
1 ... N ) ) )
11103ad2ant2 977 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  U  e.  ( CC  ^m  (
1 ... N ) ) )
12 fss 5397 . . . . . 6  |-  ( ( V : ( 1 ... N ) --> RR 
/\  RR  C_  CC )  ->  V : ( 1 ... N ) --> CC )
132, 12mpan2 652 . . . . 5  |-  ( V : ( 1 ... N ) --> RR  ->  V : ( 1 ... N ) --> CC )
145, 6elmap 6796 . . . . 5  |-  ( V  e.  ( RR  ^m  ( 1 ... N
) )  <->  V :
( 1 ... N
) --> RR )
158, 6elmap 6796 . . . . 5  |-  ( V  e.  ( CC  ^m  ( 1 ... N
) )  <->  V :
( 1 ... N
) --> CC )
1613, 14, 153imtr4i 257 . . . 4  |-  ( V  e.  ( RR  ^m  ( 1 ... N
) )  ->  V  e.  ( CC  ^m  (
1 ... N ) ) )
17163ad2ant3 978 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  V  e.  ( CC  ^m  (
1 ... N ) ) )
18 isaddrv.1 . . . 4  |-  + w  =  (  + cv `  N )
1918isaddrv 25646 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  V  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( U + w V )  =  ( x  e.  ( 1 ... N ) 
|->  ( ( U `  x )  +  ( V `  x ) ) ) )
201, 11, 17, 19syl3anc 1182 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( U + w V )  =  ( x  e.  ( 1 ... N ) 
|->  ( ( U `  x )  +  ( V `  x ) ) ) )
21 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( V : ( 1 ... N ) --> RR 
/\  x  e.  ( 1 ... N ) )  ->  ( V `  x )  e.  RR )
2221ex 423 . . . . . . . . . . . . . . 15  |-  ( V : ( 1 ... N ) --> RR  ->  ( x  e.  ( 1 ... N )  -> 
( V `  x
)  e.  RR ) )
23 ffvelrn 5663 . . . . . . . . . . . . . . . 16  |-  ( ( U : ( 1 ... N ) --> RR 
/\  x  e.  ( 1 ... N ) )  ->  ( U `  x )  e.  RR )
2423ex 423 . . . . . . . . . . . . . . 15  |-  ( U : ( 1 ... N ) --> RR  ->  ( x  e.  ( 1 ... N )  -> 
( U `  x
)  e.  RR ) )
2522, 24im2anan9 808 . . . . . . . . . . . . . 14  |-  ( ( V : ( 1 ... N ) --> RR 
/\  U : ( 1 ... N ) --> RR )  ->  (
( x  e.  ( 1 ... N )  /\  x  e.  ( 1 ... N ) )  ->  ( ( V `  x )  e.  RR  /\  ( U `
 x )  e.  RR ) ) )
26 readdcl 8820 . . . . . . . . . . . . . . 15  |-  ( ( ( U `  x
)  e.  RR  /\  ( V `  x )  e.  RR )  -> 
( ( U `  x )  +  ( V `  x ) )  e.  RR )
2726ancoms 439 . . . . . . . . . . . . . 14  |-  ( ( ( V `  x
)  e.  RR  /\  ( U `  x )  e.  RR )  -> 
( ( U `  x )  +  ( V `  x ) )  e.  RR )
2825, 27syl6 29 . . . . . . . . . . . . 13  |-  ( ( V : ( 1 ... N ) --> RR 
/\  U : ( 1 ... N ) --> RR )  ->  (
( x  e.  ( 1 ... N )  /\  x  e.  ( 1 ... N ) )  ->  ( ( U `  x )  +  ( V `  x ) )  e.  RR ) )
2928exp3acom3r 1360 . . . . . . . . . . . 12  |-  ( x  e.  ( 1 ... N )  ->  (
x  e.  ( 1 ... N )  -> 
( ( V :
( 1 ... N
) --> RR  /\  U : ( 1 ... N ) --> RR )  ->  ( ( U `
 x )  +  ( V `  x
) )  e.  RR ) ) )
3029pm2.43i 43 . . . . . . . . . . 11  |-  ( x  e.  ( 1 ... N )  ->  (
( V : ( 1 ... N ) --> RR  /\  U :
( 1 ... N
) --> RR )  -> 
( ( U `  x )  +  ( V `  x ) )  e.  RR ) )
3130exp3acom3r 1360 . . . . . . . . . 10  |-  ( V : ( 1 ... N ) --> RR  ->  ( U : ( 1 ... N ) --> RR 
->  ( x  e.  ( 1 ... N )  ->  ( ( U `
 x )  +  ( V `  x
) )  e.  RR ) ) )
3214, 31sylbi 187 . . . . . . . . 9  |-  ( V  e.  ( RR  ^m  ( 1 ... N
) )  ->  ( U : ( 1 ... N ) --> RR  ->  ( x  e.  ( 1 ... N )  -> 
( ( U `  x )  +  ( V `  x ) )  e.  RR ) ) )
3332com12 27 . . . . . . . 8  |-  ( U : ( 1 ... N ) --> RR  ->  ( V  e.  ( RR 
^m  ( 1 ... N ) )  -> 
( x  e.  ( 1 ... N )  ->  ( ( U `
 x )  +  ( V `  x
) )  e.  RR ) ) )
347, 33sylbi 187 . . . . . . 7  |-  ( U  e.  ( RR  ^m  ( 1 ... N
) )  ->  ( V  e.  ( RR  ^m  ( 1 ... N
) )  ->  (
x  e.  ( 1 ... N )  -> 
( ( U `  x )  +  ( V `  x ) )  e.  RR ) ) )
3534imp 418 . . . . . 6  |-  ( ( U  e.  ( RR 
^m  ( 1 ... N ) )  /\  V  e.  ( RR  ^m  ( 1 ... N
) ) )  -> 
( x  e.  ( 1 ... N )  ->  ( ( U `
 x )  +  ( V `  x
) )  e.  RR ) )
3635ralrimiv 2625 . . . . 5  |-  ( ( U  e.  ( RR 
^m  ( 1 ... N ) )  /\  V  e.  ( RR  ^m  ( 1 ... N
) ) )  ->  A. x  e.  (
1 ... N ) ( ( U `  x
)  +  ( V `
 x ) )  e.  RR )
37363adant1 973 . . . 4  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  A. x  e.  ( 1 ... N
) ( ( U `
 x )  +  ( V `  x
) )  e.  RR )
38 eqid 2283 . . . . 5  |-  ( x  e.  ( 1 ... N )  |->  ( ( U `  x )  +  ( V `  x ) ) )  =  ( x  e.  ( 1 ... N
)  |->  ( ( U `
 x )  +  ( V `  x
) ) )
3938fmpt 5681 . . . 4  |-  ( A. x  e.  ( 1 ... N ) ( ( U `  x
)  +  ( V `
 x ) )  e.  RR  <->  ( x  e.  ( 1 ... N
)  |->  ( ( U `
 x )  +  ( V `  x
) ) ) : ( 1 ... N
) --> RR )
4037, 39sylib 188 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  |->  ( ( U `
 x )  +  ( V `  x
) ) ) : ( 1 ... N
) --> RR )
415, 6elmap 6796 . . 3  |-  ( ( x  e.  ( 1 ... N )  |->  ( ( U `  x
)  +  ( V `
 x ) ) )  e.  ( RR 
^m  ( 1 ... N ) )  <->  ( x  e.  ( 1 ... N
)  |->  ( ( U `
 x )  +  ( V `  x
) ) ) : ( 1 ... N
) --> RR )
4240, 41sylibr 203 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  |->  ( ( U `
 x )  +  ( V `  x
) ) )  e.  ( RR  ^m  (
1 ... N ) ) )
4320, 42eqeltrd 2357 1  |-  ( ( N  e.  NN  /\  U  e.  ( RR  ^m  ( 1 ... N
) )  /\  V  e.  ( RR  ^m  (
1 ... N ) ) )  ->  ( U + w V )  e.  ( RR  ^m  (
1 ... N ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152    e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   RRcr 8736   1c1 8738    + caddc 8740   NNcn 9746   ...cfz 10782    + cvcplcv 25644
This theorem is referenced by:  negveudr  25669
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-addrcl 8798
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-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-map 6774  df-addcv 25645
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