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Theorem isdivcv2 25693
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 5875 . 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 5883 . . . . 5  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
6 cnex 8818 . . . . . 6  |-  CC  e.  _V
7 difexg 4162 . . . . . 6  |-  ( CC  e.  _V  ->  ( CC  \  { 0 } )  e.  _V )
86, 7ax-mp 8 . . . . 5  |-  ( CC 
\  { 0 } )  e.  _V
95, 8mpt2ex 6198 . . . 4  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  s  e.  ( CC  \  { 0 } ) 
|->  ( ( 1  / 
s ) ( . cv `  N ) u ) )  e. 
_V
10 oveq2 5866 . . . . . . 7  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
1110oveq2d 5874 . . . . . 6  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
12 eqidd 2284 . . . . . 6  |-  ( n  =  N  ->  ( CC  \  { 0 } )  =  ( CC 
\  { 0 } ) )
13 fveq2 5525 . . . . . . 7  |-  ( n  =  N  ->  ( . cv `  n )  =  ( . cv `  N ) )
1413oveqd 5875 . . . . . 6  |-  ( n  =  N  ->  (
( 1  /  s
) ( . cv `  n ) u )  =  ( ( 1  /  s ) ( . cv `  N
) u ) )
1511, 12, 14mpt2eq123dv 5910 . . . . 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 25692 . . . . 5  |-  / cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  (
1 ... n ) ) ,  s  e.  ( CC  \  { 0 } )  |->  ( ( 1  /  s ) ( . cv `  n
) u ) ) )
1715, 16fvmptg 5600 . . . 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 5875 . 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 5883 . . . 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 2287 . . . . . 6  |-  ( . cv `  N )  =  . t
2625a1i 10 . . . . 5  |-  ( ( u  =  U  /\  s  =  S )  ->  ( . cv `  N
)  =  . t
)
27 oveq2 5866 . . . . . 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 5879 . . . 4  |-  ( ( u  =  U  /\  s  =  S )  ->  ( ( 1  / 
s ) ( . cv `  N ) u )  =  ( ( 1  /  S
) . t U
) )
31 eqid 2283 . . . 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 5977 . . 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 2319 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149   {csn 3640   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860    ^m cmap 6772   CCcc 8735   0cc0 8737   1c1 8738    / cdiv 9423   NNcn 9746   ...cfz 10782   . cvcsmcv 25679   / cvcdivcv 25691
This theorem is referenced by:  divclcvd  25694  divclrvd  25695
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-divcv 25692
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