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Theorem isaddrv 25646
Description: Addition of complex vectors. Experimental. (Contributed by FL, 14-Sep-2013.)
Hypothesis
Ref Expression
isaddrv.1  |-  + w  =  (  + cv `  N )
Assertion
Ref Expression
isaddrv  |-  ( ( 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 ) ) ) )
Distinct variable groups:    x, U    x, N    x, V
Allowed substitution hint:    + w( x)

Proof of Theorem isaddrv
Dummy variables  v  u  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isaddrv.1 . . . . 5  |-  + w  =  (  + cv `  N )
2 oveq2 5866 . . . . . . . 8  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
32oveq2d 5874 . . . . . . 7  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
4 eqidd 2284 . . . . . . . 8  |-  ( n  =  N  ->  (
( u `  x
)  +  ( v `
 x ) )  =  ( ( u `
 x )  +  ( v `  x
) ) )
52, 4mpteq12dv 4098 . . . . . . 7  |-  ( n  =  N  ->  (
x  e.  ( 1 ... n )  |->  ( ( u `  x
)  +  ( v `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( ( u `  x )  +  ( v `  x ) ) ) )
63, 3, 5mpt2eq123dv 5910 . . . . . 6  |-  ( n  =  N  ->  (
u  e.  ( CC 
^m  ( 1 ... n ) ) ,  v  e.  ( CC 
^m  ( 1 ... n ) )  |->  ( x  e.  ( 1 ... n )  |->  ( ( u `  x
)  +  ( v `
 x ) ) ) )  =  ( u  e.  ( CC 
^m  ( 1 ... N ) ) ,  v  e.  ( CC 
^m  ( 1 ... N ) )  |->  ( x  e.  ( 1 ... N )  |->  ( ( u `  x
)  +  ( v `
 x ) ) ) ) )
7 df-addcv 25645 . . . . . 6  |-  + cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  (
1 ... n ) ) ,  v  e.  ( CC  ^m  ( 1 ... n ) ) 
|->  ( x  e.  ( 1 ... n ) 
|->  ( ( u `  x )  +  ( v `  x ) ) ) ) )
8 ovex 5883 . . . . . . 7  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
98, 8mpt2ex 6198 . . . . . 6  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( x  e.  ( 1 ... N )  |->  ( ( u `  x )  +  ( v `  x ) ) ) )  e.  _V
106, 7, 9fvmpt 5602 . . . . 5  |-  ( N  e.  NN  ->  (  + cv `  N )  =  ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( ( u `  x )  +  ( v `  x ) ) ) ) )
111, 10syl5eq 2327 . . . 4  |-  ( N  e.  NN  ->  + w  =  ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( ( u `  x )  +  ( v `  x ) ) ) ) )
12113ad2ant1 976 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  V  e.  ( CC  ^m  (
1 ... N ) ) )  ->  + w  =  ( u  e.  ( CC  ^m  ( 1 ... N ) ) ,  v  e.  ( CC 
^m  ( 1 ... N ) )  |->  ( x  e.  ( 1 ... N )  |->  ( ( u `  x
)  +  ( v `
 x ) ) ) ) )
1312oveqd 5875 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  V  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( U + w V )  =  ( U ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( x  e.  ( 1 ... N )  |->  ( ( u `  x )  +  ( v `  x ) ) ) ) V ) )
14 fveq1 5524 . . . . . 6  |-  ( u  =  U  ->  (
u `  x )  =  ( U `  x ) )
1514oveq1d 5873 . . . . 5  |-  ( u  =  U  ->  (
( u `  x
)  +  ( v `
 x ) )  =  ( ( U `
 x )  +  ( v `  x
) ) )
1615mpteq2dv 4107 . . . 4  |-  ( u  =  U  ->  (
x  e.  ( 1 ... N )  |->  ( ( u `  x
)  +  ( v `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( ( U `  x )  +  ( v `  x ) ) ) )
17 fveq1 5524 . . . . . 6  |-  ( v  =  V  ->  (
v `  x )  =  ( V `  x ) )
1817oveq2d 5874 . . . . 5  |-  ( v  =  V  ->  (
( U `  x
)  +  ( v `
 x ) )  =  ( ( U `
 x )  +  ( V `  x
) ) )
1918mpteq2dv 4107 . . . 4  |-  ( v  =  V  ->  (
x  e.  ( 1 ... N )  |->  ( ( U `  x
)  +  ( v `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( ( U `  x )  +  ( V `  x ) ) ) )
20 eqid 2283 . . . 4  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( x  e.  ( 1 ... N )  |->  ( ( u `  x )  +  ( v `  x ) ) ) )  =  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( x  e.  ( 1 ... N )  |->  ( ( u `  x )  +  ( v `  x ) ) ) )
21 ovex 5883 . . . . 5  |-  ( 1 ... N )  e. 
_V
2221mptex 5746 . . . 4  |-  ( x  e.  ( 1 ... N )  |->  ( ( U `  x )  +  ( V `  x ) ) )  e.  _V
2316, 19, 20, 22ovmpt2 5983 . . 3  |-  ( ( U  e.  ( CC 
^m  ( 1 ... N ) )  /\  V  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( U ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( x  e.  ( 1 ... N )  |->  ( ( u `  x )  +  ( v `  x ) ) ) ) V )  =  ( x  e.  ( 1 ... N ) 
|->  ( ( U `  x )  +  ( V `  x ) ) ) )
24233adant1 973 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  V  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( U
( u  e.  ( CC  ^m  ( 1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( ( u `  x )  +  ( v `  x ) ) ) ) V )  =  ( x  e.  ( 1 ... N )  |->  ( ( U `  x )  +  ( V `  x ) ) ) )
2513, 24eqtrd 2315 1  |-  ( ( 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 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^m cmap 6772   CCcc 8735   1c1 8738    + caddc 8740   NNcn 9746   ...cfz 10782    + cvcplcv 25644
This theorem is referenced by:  claddrv  25647  claddrvr  25648  addcomv  25655  addassv  25656  addidv2  25657  cnegvex2  25660  rnegvex2  25661  issubrv  25672  distmlva  25688  distsava  25689
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
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-addcv 25645
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