Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  issubcv Unicode version

Theorem issubcv 25670
Description: Substraction of complex vectors in a space of dimension  n. (Contributed by FL, 15-Sep-2013.)
Hypotheses
Ref Expression
issubcv.1  |-  + w  =  (  + cv `  N )
issubcv.2  |-  - w  =  (  - cv  `  N
)
Assertion
Ref Expression
issubcv  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  V  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( U - w V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V + w
w )  =  U ) )
Distinct variable groups:    w, U    w, N    w, V    w, + w
Allowed substitution hint:    - w( w)

Proof of Theorem issubcv
Dummy variables  v  u  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . . . 7  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21oveq2d 5874 . . . . . 6  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
3 fveq2 5525 . . . . . . . . 9  |-  ( n  =  N  ->  (  + cv `  n )  =  (  + cv `  N ) )
43oveqd 5875 . . . . . . . 8  |-  ( n  =  N  ->  (
v (  + cv `  n ) w )  =  ( v (  + cv `  N
) w ) )
54eqeq1d 2291 . . . . . . 7  |-  ( n  =  N  ->  (
( v (  + cv `  n ) w )  =  u  <-> 
( v (  + cv `  N ) w )  =  u ) )
62, 5riotaeqbidv 6307 . . . . . 6  |-  ( n  =  N  ->  ( iota_ w  e.  ( CC 
^m  ( 1 ... n ) ) ( v (  + cv `  n ) w )  =  u )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) )
72, 2, 6mpt2eq123dv 5910 . . . . 5  |-  ( n  =  N  ->  (
u  e.  ( CC 
^m  ( 1 ... n ) ) ,  v  e.  ( CC 
^m  ( 1 ... n ) )  |->  (
iota_ w  e.  ( CC  ^m  ( 1 ... n ) ) ( v (  + cv `  n ) w )  =  u ) )  =  ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) )
8 df-subcatv 25665 . . . . 5  |-  - cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  (
1 ... n ) ) ,  v  e.  ( CC  ^m  ( 1 ... n ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... n ) ) ( v (  + cv `  n ) w )  =  u ) ) )
9 ovex 5883 . . . . . 6  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
109, 9mpt2ex 6198 . . . . 5  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( v (  + cv `  N
) w )  =  u ) )  e. 
_V
117, 8, 10fvmpt 5602 . . . 4  |-  ( N  e.  NN  ->  (  - cv  `  N )  =  ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) )
12 riotaex 6308 . . . . . 6  |-  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( V (  + cv `  N
) w )  =  U )  e.  _V
13 eqeq2 2292 . . . . . . . . 9  |-  ( u  =  U  ->  (
( v (  + cv `  N ) w )  =  u  <-> 
( v (  + cv `  N ) w )  =  U ) )
1413riotabidv 6306 . . . . . . . 8  |-  ( u  =  U  ->  ( iota_ w  e.  ( CC 
^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  U ) )
15 oveq1 5865 . . . . . . . . . 10  |-  ( v  =  V  ->  (
v (  + cv `  N ) w )  =  ( V (  + cv `  N
) w ) )
1615eqeq1d 2291 . . . . . . . . 9  |-  ( v  =  V  ->  (
( v (  + cv `  N ) w )  =  U  <-> 
( V (  + cv `  N ) w )  =  U ) )
1716riotabidv 6306 . . . . . . . 8  |-  ( v  =  V  ->  ( iota_ w  e.  ( CC 
^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  U )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) )
18 eqid 2283 . . . . . . . 8  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( v (  + cv `  N
) w )  =  u ) )  =  ( u  e.  ( CC  ^m  ( 1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) )
1914, 17, 18ovmpt2g 5982 . . . . . . 7  |-  ( ( U  e.  ( CC 
^m  ( 1 ... N ) )  /\  V  e.  ( CC  ^m  ( 1 ... N
) )  /\  ( iota_ w  e.  ( CC 
^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U )  e. 
_V )  ->  ( U ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) V )  =  ( iota_ w  e.  ( CC  ^m  (
1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) )
20193exp 1150 . . . . . 6  |-  ( U  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( V  e.  ( CC  ^m  ( 1 ... N
) )  ->  (
( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U )  e.  _V  ->  ( U ( u  e.  ( CC  ^m  (
1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) V )  =  ( iota_ w  e.  ( CC  ^m  (
1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) ) )
2112, 20mpii 39 . . . . 5  |-  ( 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 ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) V )  =  ( iota_ w  e.  ( CC  ^m  (
1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) )
22 oveq 5864 . . . . . . 7  |-  ( (  - cv  `  N
)  =  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( v (  + cv `  N
) w )  =  u ) )  -> 
( U (  - cv  `  N ) V )  =  ( U ( u  e.  ( CC  ^m  ( 1 ... N ) ) ,  v  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) V ) )
2322eqeq1d 2291 . . . . . 6  |-  ( (  - cv  `  N
)  =  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( v (  + cv `  N
) w )  =  u ) )  -> 
( ( U (  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U )  <->  ( U ( u  e.  ( CC 
^m  ( 1 ... N ) ) ,  v  e.  ( CC 
^m  ( 1 ... N ) )  |->  (
iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) V )  =  (
iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) )
2423imbi2d 307 . . . . 5  |-  ( (  - cv  `  N
)  =  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( v (  + cv `  N
) w )  =  u ) )  -> 
( ( V  e.  ( CC  ^m  (
1 ... N ) )  ->  ( U (  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) )  <->  ( V  e.  ( CC  ^m  (
1 ... N ) )  ->  ( U ( u  e.  ( CC 
^m  ( 1 ... N ) ) ,  v  e.  ( CC 
^m  ( 1 ... N ) )  |->  (
iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( v (  + cv `  N ) w )  =  u ) ) V )  =  (
iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) ) )
2521, 24syl5ibr 212 . . . 4  |-  ( (  - cv  `  N
)  =  ( u  e.  ( CC  ^m  ( 1 ... N
) ) ,  v  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( v (  + cv `  N
) w )  =  u ) )  -> 
( U  e.  ( CC  ^m  ( 1 ... N ) )  ->  ( V  e.  ( CC  ^m  (
1 ... N ) )  ->  ( U (  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) ) )
2611, 25syl 15 . . 3  |-  ( N  e.  NN  ->  ( U  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( V  e.  ( CC  ^m  ( 1 ... N
) )  ->  ( U (  - cv  `  N ) V )  =  ( iota_ w  e.  ( CC  ^m  (
1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) ) )
27263imp 1145 . 2  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  V  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( U
(  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) )
28 issubcv.1 . . 3  |-  + w  =  (  + cv `  N )
29 oveq 5864 . . . . . . . 8  |-  ( + w  =  (  + cv `  N )  ->  ( V + w w )  =  ( V (  + cv `  N ) w ) )
3029adantr 451 . . . . . . 7  |-  ( ( + w  =  (  + cv `  N
)  /\  w  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( V + w w )  =  ( V (  + cv `  N ) w ) )
3130eqeq1d 2291 . . . . . 6  |-  ( ( + w  =  (  + cv `  N
)  /\  w  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( ( V + w w )  =  U  <->  ( V
(  + cv `  N
) w )  =  U ) )
3231riotabidva 6321 . . . . 5  |-  ( + w  =  (  + cv `  N )  ->  ( iota_ w  e.  ( CC  ^m  (
1 ... N ) ) ( V + w
w )  =  U )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( V (  + cv `  N
) w )  =  U ) )
3332eqeq2d 2294 . . . 4  |-  ( + w  =  (  + cv `  N )  ->  ( ( U - w V )  =  ( iota_ w  e.  ( CC  ^m  (
1 ... N ) ) ( V + w
w )  =  U )  <->  ( U - w V )  =  (
iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) )
34 issubcv.2 . . . . 5  |-  - w  =  (  - cv  `  N
)
35 oveq 5864 . . . . . 6  |-  ( - w  =  (  - cv  `  N )  -> 
( U - w V )  =  ( U (  - cv  `  N ) V ) )
3635eqeq1d 2291 . . . . 5  |-  ( - w  =  (  - cv  `  N )  -> 
( ( U - w V )  =  (
iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U )  <->  ( U
(  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) )
3734, 36ax-mp 8 . . . 4  |-  ( ( U - w V
)  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( V (  + cv `  N
) w )  =  U )  <->  ( U
(  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) )
3833, 37syl6bb 252 . . 3  |-  ( + w  =  (  + cv `  N )  ->  ( ( U - w V )  =  ( iota_ w  e.  ( CC  ^m  (
1 ... N ) ) ( V + w
w )  =  U )  <->  ( U (  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) ) )
3928, 38ax-mp 8 . 2  |-  ( ( U - w V
)  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( V + w w )  =  U )  <->  ( U
(  - cv  `  N
) V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V (  + cv `  N ) w )  =  U ) )
4027, 39sylibr 203 1  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) )  /\  V  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( U - w V )  =  ( iota_ w  e.  ( CC  ^m  ( 1 ... N ) ) ( V + w
w )  =  U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   iota_crio 6297    ^m cmap 6772   CCcc 8735   1c1 8738   NNcn 9746   ...cfz 10782    + cvcplcv 25644    - cv cmcv 25664
This theorem is referenced by:  subaddv  25671  subclcvd  25673
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-riota 6304  df-subcatv 25665
  Copyright terms: Public domain W3C validator