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Theorem isucv 25677
Description: Negative of a complex vector. (Contributed by FL, 15-Sep-2013.)
Hypotheses
Ref Expression
isucv.1  |-  ~ w  =  ( - cv `  N )
isucv.2  |-  0 w  =  ( 0 cv
`  N )
isucv.3  |-  - w  =  (  - cv  `  N
)
Assertion
Ref Expression
isucv  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ~ w `  U )  =  ( 0 w - w U ) )

Proof of Theorem isucv
Dummy variables  u  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isucv.1 . 2  |-  ~ w  =  ( - cv `  N )
2 isucv.2 . . . 4  |-  0 w  =  ( 0 cv
`  N )
3 isucv.3 . . . . . 6  |-  - w  =  (  - cv  `  N
)
4 simpl 443 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  N  e.  NN )
5 ovex 5883 . . . . . . . . . . 11  |-  ( CC 
^m  ( 1 ... N ) )  e. 
_V
65mptex 5746 . . . . . . . . . 10  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) )  e.  _V
7 oveq2 5866 . . . . . . . . . . . . 13  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
87oveq2d 5874 . . . . . . . . . . . 12  |-  ( n  =  N  ->  ( CC  ^m  ( 1 ... n ) )  =  ( CC  ^m  (
1 ... N ) ) )
9 fveq2 5525 . . . . . . . . . . . . 13  |-  ( n  =  N  ->  (  - cv  `  n )  =  (  - cv  `  N ) )
10 fveq2 5525 . . . . . . . . . . . . 13  |-  ( n  =  N  ->  (
0 cv `  n
)  =  ( 0 cv `  N ) )
11 eqidd 2284 . . . . . . . . . . . . 13  |-  ( n  =  N  ->  u  =  u )
129, 10, 11oveq123d 5879 . . . . . . . . . . . 12  |-  ( n  =  N  ->  (
( 0 cv `  n
) (  - cv  `  n ) u )  =  ( ( 0 cv `  N ) (  - cv  `  N
) u ) )
138, 12mpteq12dv 4098 . . . . . . . . . . 11  |-  ( n  =  N  ->  (
u  e.  ( CC 
^m  ( 1 ... n ) )  |->  ( ( 0 cv `  n
) (  - cv  `  n ) u ) )  =  ( u  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) ) )
14 df-ucv 25676 . . . . . . . . . . 11  |-  - cv  =  ( n  e.  NN  |->  ( u  e.  ( CC  ^m  (
1 ... n ) ) 
|->  ( ( 0 cv
`  n ) (  - cv  `  n
) u ) ) )
1513, 14fvmptg 5600 . . . . . . . . . 10  |-  ( ( N  e.  NN  /\  ( u  e.  ( CC  ^m  ( 1 ... N ) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) )  e.  _V )  ->  ( - cv `  N
)  =  ( u  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) ) )
164, 6, 15sylancl 643 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( - cv `  N
)  =  ( u  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) ) )
1716fveq1d 5527 . . . . . . . 8  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ( - cv `  N ) `  U
)  =  ( ( u  e.  ( CC 
^m  ( 1 ... N ) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) ) `  U ) )
18 simpr 447 . . . . . . . . 9  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  ->  U  e.  ( CC  ^m  ( 1 ... N
) ) )
19 ovex 5883 . . . . . . . . 9  |-  ( ( 0 cv `  N
) (  - cv  `  N ) U )  e.  _V
20 oveq2 5866 . . . . . . . . . 10  |-  ( u  =  U  ->  (
( 0 cv `  N
) (  - cv  `  N ) u )  =  ( ( 0 cv `  N ) (  - cv  `  N
) U ) )
21 eqid 2283 . . . . . . . . . 10  |-  ( u  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) )  =  ( u  e.  ( CC  ^m  ( 1 ... N
) )  |->  ( ( 0 cv `  N
) (  - cv  `  N ) u ) )
2220, 21fvmptg 5600 . . . . . . . . 9  |-  ( ( U  e.  ( CC 
^m  ( 1 ... N ) )  /\  ( ( 0 cv
`  N ) (  - cv  `  N
) U )  e. 
_V )  ->  (
( u  e.  ( CC  ^m  ( 1 ... N ) ) 
|->  ( ( 0 cv
`  N ) (  - cv  `  N
) u ) ) `
 U )  =  ( ( 0 cv
`  N ) (  - cv  `  N
) U ) )
2318, 19, 22sylancl 643 . . . . . . . 8  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ( u  e.  ( CC  ^m  (
1 ... N ) ) 
|->  ( ( 0 cv
`  N ) (  - cv  `  N
) u ) ) `
 U )  =  ( ( 0 cv
`  N ) (  - cv  `  N
) U ) )
2417, 23eqtrd 2315 . . . . . . 7  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ( - cv `  N ) `  U
)  =  ( ( 0 cv `  N
) (  - cv  `  N ) U ) )
25 oveq 5864 . . . . . . . 8  |-  ( - w  =  (  - cv  `  N )  -> 
( ( 0 cv
`  N ) - w U )  =  ( ( 0 cv `  N
) (  - cv  `  N ) U ) )
2625eqeq2d 2294 . . . . . . 7  |-  ( - w  =  (  - cv  `  N )  -> 
( ( ( - cv `  N ) `  U )  =  ( ( 0 cv `  N
) - w U
)  <->  ( ( - cv `  N ) `  U )  =  ( ( 0 cv `  N
) (  - cv  `  N ) U ) ) )
2724, 26syl5ibr 212 . . . . . 6  |-  ( - w  =  (  - cv  `  N )  -> 
( ( N  e.  NN  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( ( - cv `  N ) `
 U )  =  ( ( 0 cv
`  N ) - w U ) ) )
283, 27ax-mp 8 . . . . 5  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ( - cv `  N ) `  U
)  =  ( ( 0 cv `  N
) - w U
) )
29 oveq1 5865 . . . . . 6  |-  ( 0 w  =  ( 0 cv `  N )  ->  ( 0 w - w U )  =  ( ( 0 cv
`  N ) - w U ) )
3029eqeq2d 2294 . . . . 5  |-  ( 0 w  =  ( 0 cv `  N )  ->  ( ( (
- cv `  N
) `  U )  =  ( 0 w - w U )  <->  ( ( - cv `  N ) `
 U )  =  ( ( 0 cv
`  N ) - w U ) ) )
3128, 30syl5ibr 212 . . . 4  |-  ( 0 w  =  ( 0 cv `  N )  ->  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( ( - cv `  N ) `
 U )  =  ( 0 w - w U ) ) )
322, 31ax-mp 8 . . 3  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ( - cv `  N ) `  U
)  =  ( 0 w - w U
) )
33 fveq1 5524 . . . 4  |-  ( ~ w  =  ( - cv `  N )  -> 
( ~ w `  U )  =  ( ( - cv `  N
) `  U )
)
3433eqeq1d 2291 . . 3  |-  ( ~ w  =  ( - cv `  N )  -> 
( ( ~ w `  U )  =  ( 0 w - w U )  <->  ( ( - cv `  N ) `
 U )  =  ( 0 w - w U ) ) )
3532, 34syl5ibr 212 . 2  |-  ( ~ w  =  ( - cv `  N )  -> 
( ( N  e.  NN  /\  U  e.  ( CC  ^m  (
1 ... N ) ) )  ->  ( ~ w `  U )  =  ( 0 w - w U ) ) )
361, 35ax-mp 8 1  |-  ( ( N  e.  NN  /\  U  e.  ( CC  ^m  ( 1 ... N
) ) )  -> 
( ~ w `  U )  =  ( 0 w - w U ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   CCcc 8735   1c1 8738   NNcn 9746   ...cfz 10782   0 cvc0cv 25650    - cv cmcv 25664   - cvcnegcv 25675
This theorem is referenced by:  isucvr  25678
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-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-pr 4214
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-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-ucv 25676
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