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Theorem sigarval 27707
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 444 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21fveq2d 5691 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( * `  x
)  =  ( * `
 A ) )
3 simpr 448 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
42, 3oveq12d 6058 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( * `  x )  x.  y
)  =  ( ( * `  A )  x.  B ) )
54fveq2d 5691 . 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 5701 . 2  |-  ( Im
`  ( ( * `
 A )  x.  B ) )  e. 
_V
85, 6, 7ovmpt2a 6163 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A G B )  =  ( Im
`  ( ( * `
 A )  x.  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   CCcc 8944    x. cmul 8951   *ccj 11856   Imcim 11858
This theorem is referenced by:  sigarim  27708  sigarac  27709  sigaraf  27710  sigarmf  27711  sigarls  27714  sigarid  27715  sigardiv  27718  sharhght  27722
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045
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