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Theorem ismulcv 25784
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 8834 . . . . . . 7  |-  CC  e.  _V
4 ovex 5899 . . . . . . 7  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
53, 4mpt2ex 6214 . . . . . 6  |-  ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... N ) ) 
|->  ( x  e.  ( 1 ... N ) 
|->  ( s  x.  (
u `  x )
) ) )  e. 
_V
6 eqidd 2297 . . . . . . . 8  |-  ( n  =  N  ->  CC  =  CC )
7 oveq2 5882 . . . . . . . . 9  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
87oveq2d 5890 . . . . . . . 8  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
9 eqidd 2297 . . . . . . . . 9  |-  ( n  =  N  ->  (
s  x.  ( u `
 x ) )  =  ( s  x.  ( u `  x
) ) )
107, 9mpteq12dv 4114 . . . . . . . 8  |-  ( n  =  N  ->  (
x  e.  ( 1 ... n )  |->  ( s  x.  ( u `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( s  x.  ( u `  x ) ) ) )
116, 8, 10mpt2eq123dv 5926 . . . . . . 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 25783 . . . . . . 7  |-  . cv  =  ( n  e.  NN  |->  ( s  e.  CC ,  u  e.  ( CC  ^m  (
1 ... n ) ) 
|->  ( x  e.  ( 1 ... n ) 
|->  ( s  x.  (
u `  x )
) ) ) )
1311, 12fvmptg 5616 . . . . . 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 5891 . . . 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 5899 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
1918mptex 5762 . . . . . 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 5881 . . . . . . 7  |-  ( s  =  S  ->  (
s  x.  ( u `
 x ) )  =  ( S  x.  ( u `  x
) ) )
2221mpteq2dv 4123 . . . . . 6  |-  ( s  =  S  ->  (
x  e.  ( 1 ... N )  |->  ( s  x.  ( u `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( S  x.  ( u `  x ) ) ) )
23 fveq1 5540 . . . . . . . 8  |-  ( u  =  U  ->  (
u `  x )  =  ( U `  x ) )
2423oveq2d 5890 . . . . . . 7  |-  ( u  =  U  ->  ( S  x.  ( u `  x ) )  =  ( S  x.  ( U `  x )
) )
2524mpteq2dv 4123 . . . . . 6  |-  ( u  =  U  ->  (
x  e.  ( 1 ... N )  |->  ( S  x.  ( u `
 x ) ) )  =  ( x  e.  ( 1 ... N )  |->  ( S  x.  ( U `  x ) ) ) )
26 eqid 2296 . . . . . 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 5998 . . . . 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 2328 . . 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 5880 . . . 4  |-  ( . t  =  ( . cv `  N )  ->  ( S . t U )  =  ( S ( . cv `  N ) U ) )
3130eqeq1d 2304 . . 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 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788   CCcc 8751   1c1 8754    x. cmul 8758   NNcn 9762   ...cfz 10798   . cvcsmcv 25782
This theorem is referenced by:  clsmulcv  25785  clsmulrv  25786  mulone  25788  mulmulvec  25790  distmlva  25791  distsava  25792
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-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-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-mulcv 25783
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