Users' Mathboxes Mathbox for Saveliy Skresanov < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sigarval Structured version   Unicode version

Theorem sigarval 29884
Description: Define the signed area by treating complex numbers as vectors with two components. (Contributed by Saveliy Skresanov, 19-Sep-2017.)
Hypothesis
Ref Expression
sigar  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
Assertion
Ref Expression
sigarval  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
Distinct variable groups:    x, y, A    x, B, y
Allowed substitution hints:    G( x, y)

Proof of Theorem sigarval
StepHypRef Expression
1 simpl 457 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21fveq2d 5694 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( * `  x
)  =  ( * `
 A ) )
3 simpr 461 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
42, 3oveq12d 6108 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( * `  x )  x.  y
)  =  ( ( * `  A )  x.  B ) )
54fveq2d 5694 . 2  |-  ( ( x  =  A  /\  y  =  B )  ->  ( Im `  (
( * `  x
)  x.  y ) )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
6 sigar . 2  |-  G  =  ( x  e.  CC ,  y  e.  CC  |->  ( Im `  ( ( * `  x )  x.  y ) ) )
7 fvex 5700 . 2  |-  ( Im
`  ( ( * `
 A )  x.  B ) )  e. 
_V
85, 6, 7ovmpt2a 6220 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   ` cfv 5417  (class class class)co 6090    e. cmpt2 6092   CCcc 9279    x. cmul 9286   *ccj 12584   Imcim 12586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095
This theorem is referenced by:  sigarim  29885  sigarac  29886  sigaraf  29887  sigarmf  29888  sigarls  29891  sigarid  29892  sigardiv  29895  sharhght  29899
  Copyright terms: Public domain W3C validator