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Theorem issubcv 25773
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 5882 . . . . . . 7  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
21oveq2d 5890 . . . . . 6  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
3 fveq2 5541 . . . . . . . . 9  |-  ( n  =  N  ->  (  + cv `  n )  =  (  + cv `  N ) )
43oveqd 5891 . . . . . . . 8  |-  ( n  =  N  ->  (
v (  + cv `  n ) w )  =  ( v (  + cv `  N
) w ) )
54eqeq1d 2304 . . . . . . 7  |-  ( n  =  N  ->  (
( v (  + cv `  n ) w )  =  u  <-> 
( v (  + cv `  N ) w )  =  u ) )
62, 5riotaeqbidv 6323 . . . . . 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 5926 . . . . 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 25768 . . . . 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 5899 . . . . . 6  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
109, 9mpt2ex 6214 . . . . 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 5618 . . . 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 6324 . . . . . 6  |-  ( iota_ w  e.  ( CC  ^m  ( 1 ... N
) ) ( V (  + cv `  N
) w )  =  U )  e.  _V
13 eqeq2 2305 . . . . . . . . 9  |-  ( u  =  U  ->  (
( v (  + cv `  N ) w )  =  u  <-> 
( v (  + cv `  N ) w )  =  U ) )
1413riotabidv 6322 . . . . . . . 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 5881 . . . . . . . . . 10  |-  ( v  =  V  ->  (
v (  + cv `  N ) w )  =  ( V (  + cv `  N
) w ) )
1615eqeq1d 2304 . . . . . . . . 9  |-  ( v  =  V  ->  (
( v (  + cv `  N ) w )  =  U  <-> 
( V (  + cv `  N ) w )  =  U ) )
1716riotabidv 6322 . . . . . . . 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 2296 . . . . . . . 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 5998 . . . . . . 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 5880 . . . . . . 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 2304 . . . . . 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 5880 . . . . . . . 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 2304 . . . . . 6  |-  ( ( + w  =  (  + cv `  N
)  /\  w  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( ( V + w w )  =  U  <->  ( V
(  + cv `  N
) w )  =  U ) )
3231riotabidva 6337 . . . . 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 2307 . . . 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 5880 . . . . . 6  |-  ( - w  =  (  - cv  `  N )  -> 
( U - w V )  =  ( U (  - cv  `  N ) V ) )
3635eqeq1d 2304 . . . . 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 1632    e. wcel 1696   _Vcvv 2801   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   iota_crio 6313    ^m cmap 6788   CCcc 8751   1c1 8754   NNcn 9762   ...cfz 10798    + cvcplcv 25747    - cv cmcv 25767
This theorem is referenced by:  subaddv  25774  subclcvd  25776
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
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-riota 6320  df-subcatv 25768
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