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Theorem ismulcv 25681
Description: Multiplication of complex vectors by a scalar in a space of dimension  n. (Contributed by FL, 15-Sep-2013.) (Revised by Mario Carneiro, 23-Aug-2014.)
Hypothesis
Ref Expression
ismulcv.1  |-  . t  =  ( . cv `  N )
Assertion
Ref Expression
ismulcv  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( S . t U )  =  ( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) )
Distinct variable groups:    x, N    x, S    x, U
Allowed substitution hint:    . t( x)

Proof of Theorem ismulcv
Dummy variables  n  s  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ismulcv.1 . 2  |-  . t  =  ( . cv `  N )
2 simp1 955 . . . . . 6  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  N  e.  NN )
3 cnex 8818 . . . . . . 7  |-  CC  e.  _V
4 ovex 5883 . . . . . . 7  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
53, 4mpt2ex 6198 . . . . . 6  |-  ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( s  x.  (
u `  x )
) ) )  e. 
_V
6 eqidd 2284 . . . . . . . 8  |-  ( n  =  N  ->  CC  =  CC )
7 oveq2 5866 . . . . . . . . 9  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
87oveq2d 5874 . . . . . . . 8  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
9 eqidd 2284 . . . . . . . . 9  |-  ( n  =  N  ->  (
s  x.  ( u `
 x ) )  =  ( s  x.  ( u `  x
) ) )
107, 9mpteq12dv 4098 . . . . . . . 8  |-  ( n  =  N  ->  (
x  e.  ( 1 ... n )  |->  ( s  x.  ( u `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( s  x.  ( u `  x ) ) ) )
116, 8, 10mpt2eq123dv 5910 . . . . . . 7  |-  ( n  =  N  ->  (
s  e.  CC ,  u  e.  ( CC  ^m  ( 1 ... n
) )  |->  ( x  e.  ( 1 ... n )  |->  ( s  x.  ( u `  x ) ) ) )  =  ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( s  x.  (
u `  x )
) ) ) )
12 df-mulcv 25680 . . . . . . 7  |-  . cv  =  ( n  e.  NN  |->  ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... n ) ) 
|->  ( x  e.  ( 1 ... n ) 
|->  ( s  x.  (
u `  x )
) ) ) )
1311, 12fvmptg 5600 . . . . . 6  |-  ( ( N  e.  NN  /\  ( s  e.  CC ,  u  e.  ( CC  ^m  ( 1 ... N ) )  |->  ( x  e.  ( 1 ... N )  |->  ( s  x.  ( u `
 x ) ) ) )  e.  _V )  ->  ( . cv `  N )  =  ( s  e.  CC ,  u  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( x  e.  ( 1 ... N )  |->  ( s  x.  ( u `  x ) ) ) ) )
142, 5, 13sylancl 643 . . . . 5  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( . cv `  N )  =  ( s  e.  CC ,  u  e.  ( CC  ^m  ( 1 ... N ) )  |->  ( x  e.  ( 1 ... N )  |->  ( s  x.  ( u `
 x ) ) ) ) )
1514oveqd 5875 . . . 4  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( S
( . cv `  N
) U )  =  ( S ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( s  x.  (
u `  x )
) ) ) U ) )
16 simp2 956 . . . . 5  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  S  e.  CC )
17 simp3 957 . . . . 5  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  U  e.  ( CC  ^m  (
1 ... N ) ) )
18 ovex 5883 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
1918mptex 5746 . . . . . 6  |-  ( x  e.  ( 1 ... N )  |->  ( S  x.  ( U `  x ) ) )  e.  _V
2019a1i 10 . . . . 5  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) )  e.  _V )
21 oveq1 5865 . . . . . . 7  |-  ( s  =  S  ->  (
s  x.  ( u `
 x ) )  =  ( S  x.  ( u `  x
) ) )
2221mpteq2dv 4107 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( 1 ... N )  |->  ( s  x.  ( u `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( S  x.  ( u `  x ) ) ) )
23 fveq1 5524 . . . . . . . 8  |-  ( u  =  U  ->  (
u `  x )  =  ( U `  x ) )
2423oveq2d 5874 . . . . . . 7  |-  ( u  =  U  ->  ( S  x.  ( u `  x ) )  =  ( S  x.  ( U `  x )
) )
2524mpteq2dv 4107 . . . . . 6  |-  ( u  =  U  ->  (
x  e.  ( 1 ... N )  |->  ( S  x.  ( u `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( S  x.  ( U `  x ) ) ) )
26 eqid 2283 . . . . . 6  |-  ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( s  x.  (
u `  x )
) ) )  =  ( s  e.  CC ,  u  e.  ( CC  ^m  ( 1 ... N ) )  |->  ( x  e.  ( 1 ... N )  |->  ( s  x.  ( u `
 x ) ) ) )
2722, 25, 26ovmpt2g 5982 . . . . 5  |-  ( ( S  e.  CC  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  (
x  e.  ( 1 ... N )  |->  ( S  x.  ( U `
 x ) ) )  e.  _V )  ->  ( S ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( s  x.  (
u `  x )
) ) ) U )  =  ( x  e.  ( 1 ... N )  |->  ( S  x.  ( U `  x ) ) ) )
2816, 17, 20, 27syl3anc 1182 . . . 4  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( S
( s  e.  CC ,  u  e.  ( CC  ^m  ( 1 ... N ) )  |->  ( x  e.  ( 1 ... N )  |->  ( s  x.  ( u `
 x ) ) ) ) U )  =  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) ) )
2915, 28eqtrd 2315 . . 3  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( S
( . cv `  N
) U )  =  ( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) )
30 oveq 5864 . . . 4  |-  ( . t  =  ( . cv `  N )  ->  ( S . t U )  =  ( S ( . cv `  N ) U ) )
3130eqeq1d 2291 . . 3  |-  ( . t  =  ( . cv `  N )  ->  ( ( S . t U )  =  ( x  e.  ( 1 ... N
)  |->  ( S  x.  ( U `  x ) ) )  <->  ( S
( . cv `  N
) U )  =  ( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) ) )
3229, 31syl5ibr 212 . 2  |-  ( . t  =  ( . cv `  N )  ->  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( S . t U )  =  ( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) ) )
331, 32ax-mp 8 1  |-  ( ( N  e.  NN  /\  S  e.  CC  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( S . t U )  =  ( x  e.  ( 1 ... N ) 
|->  ( S  x.  ( U `  x )
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^m cmap 6772   CCcc 8735   1c1 8738    x. cmul 8742   NNcn 9746   ...cfz 10782   . cvcsmcv 25679
This theorem is referenced by:  clsmulcv  25682  clsmulrv  25683  mulone  25685  mulmulvec  25687  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  ax-cnex 8793
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-mulcv 25680
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