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Theorem isdivcv2 25796
Description: Division of complex vectors by a scalar in a space of dimension  N. (Contributed by FL, 29-May-2014.)
Hypotheses
Ref Expression
isdivcv2.1  |-  / t  =  ( / cv `  N )
isdivcv2.2  |-  . t  =  ( . cv `  N )
Assertion
Ref Expression
isdivcv2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( U / t S )  =  ( ( 1  /  S
) . t U
) )

Proof of Theorem isdivcv2
Dummy variables  n  s  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isdivcv2.1 . . . 4  |-  / t  =  ( / cv `  N )
21a1i 10 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  / t  =  ( / cv `  N
) )
32oveqd 5891 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( U / t S )  =  ( U ( / cv `  N ) S ) )
4 simp1 955 . . . 4  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  N  e.  NN )
5 ovex 5899 . . . . 5  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
6 cnex 8834 . . . . . 6  |-  CC  e.  _V
7 difexg 4178 . . . . . 6  |-  ( CC  e.  _V  ->  ( CC  \  { 0 } )  e.  _V )
86, 7ax-mp 8 . . . . 5  |-  ( CC 
\  { 0 } )  e.  _V
95, 8mpt2ex 6214 . . . 4  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  s  e.  ( CC  \  { 0 } ) 
|->  ( ( 1  / 
s ) ( . cv `  N ) u ) )  e. 
_V
10 oveq2 5882 . . . . . . 7  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
1110oveq2d 5890 . . . . . 6  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
12 eqidd 2297 . . . . . 6  |-  ( n  =  N  ->  ( CC  \  { 0 } )  =  ( CC 
\  { 0 } ) )
13 fveq2 5541 . . . . . . 7  |-  ( n  =  N  ->  ( . cv `  n )  =  ( . cv `  N ) )
1413oveqd 5891 . . . . . 6  |-  ( n  =  N  ->  (
( 1  /  s
) ( . cv `  n ) u )  =  ( ( 1  /  s ) ( . cv `  N
) u ) )
1511, 12, 14mpt2eq123dv 5926 . . . . 5  |-  ( n  =  N  ->  (
u  e.  ( CC 
^m  ( 1 ... n ) ) ,  s  e.  ( CC 
\  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  n
) u ) )  =  ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  s  e.  ( CC  \  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  N
) u ) ) )
16 df-divcv 25795 . . . . 5  |-  / cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  (
1 ... n ) ) ,  s  e.  ( CC  \  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  n
) u ) ) )
1715, 16fvmptg 5616 . . . 4  |-  ( ( N  e.  NN  /\  ( u  e.  ( CC  ^m  ( 1 ... N ) ) ,  s  e.  ( CC 
\  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  N
) u ) )  e.  _V )  -> 
( / cv `  N
)  =  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  s  e.  ( CC  \  { 0 } ) 
|->  ( ( 1  / 
s ) ( . cv `  N ) u ) ) )
184, 9, 17sylancl 643 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( / cv `  N )  =  ( u  e.  ( CC 
^m  ( 1 ... N ) ) ,  s  e.  ( CC 
\  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  N
) u ) ) )
1918oveqd 5891 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( U ( / cv `  N
) S )  =  ( U ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  s  e.  ( CC  \  { 0 } ) 
|->  ( ( 1  / 
s ) ( . cv `  N ) u ) ) S ) )
20 simp2 956 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  U  e.  ( CC  ^m  ( 1 ... N ) ) )
21 simp3 957 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  S  e.  ( CC  \  { 0 } ) )
22 ovex 5899 . . . 4  |-  ( ( 1  /  S ) . t U )  e.  _V
2322a1i 10 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( ( 1  /  S ) . t U )  e. 
_V )
24 isdivcv2.2 . . . . . . 7  |-  . t  =  ( . cv `  N )
2524eqcomi 2300 . . . . . 6  |-  ( . cv `  N )  =  . t
2625a1i 10 . . . . 5  |-  ( ( u  =  U  /\  s  =  S )  ->  ( . cv `  N
)  =  . t
)
27 oveq2 5882 . . . . . 6  |-  ( s  =  S  ->  (
1  /  s )  =  ( 1  /  S ) )
2827adantl 452 . . . . 5  |-  ( ( u  =  U  /\  s  =  S )  ->  ( 1  /  s
)  =  ( 1  /  S ) )
29 simpl 443 . . . . 5  |-  ( ( u  =  U  /\  s  =  S )  ->  u  =  U )
3026, 28, 29oveq123d 5895 . . . 4  |-  ( ( u  =  U  /\  s  =  S )  ->  ( ( 1  / 
s ) ( . cv `  N ) u )  =  ( ( 1  /  S
) . t U
) )
31 eqid 2296 . . . 4  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  s  e.  ( CC  \  { 0 } ) 
|->  ( ( 1  / 
s ) ( . cv `  N ) u ) )  =  ( u  e.  ( CC  ^m  ( 1 ... N ) ) ,  s  e.  ( CC  \  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  N
) u ) )
3230, 31ovmpt2ga 5993 . . 3  |-  ( ( U  e.  ( CC 
^m  ( 1 ... N ) )  /\  S  e.  ( CC  \  { 0 } )  /\  ( ( 1  /  S ) . t U )  e. 
_V )  ->  ( U ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  s  e.  ( CC  \  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  N
) u ) ) S )  =  ( ( 1  /  S
) . t U
) )
3320, 21, 23, 32syl3anc 1182 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( U ( u  e.  ( CC 
^m  ( 1 ... N ) ) ,  s  e.  ( CC 
\  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  N
) u ) ) S )  =  ( ( 1  /  S
) . t U
) )
343, 19, 333eqtrd 2332 1  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  S  e.  ( CC  \  {
0 } ) )  ->  ( U / t S )  =  ( ( 1  /  S
) . t U
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162   {csn 3653   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788   CCcc 8751   0cc0 8753   1c1 8754    / cdiv 9439   NNcn 9762   ...cfz 10798   . cvcsmcv 25782   / cvcdivcv 25794
This theorem is referenced by:  divclcvd  25797  divclrvd  25798
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-divcv 25795
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