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Theorem sigarval 27816
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 5732 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( * `  x
)  =  ( * `
 A ) )
3 simpr 448 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
42, 3oveq12d 6099 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( * `  x )  x.  y
)  =  ( ( * `  A )  x.  B ) )
54fveq2d 5732 . 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 5742 . 2  |-  ( Im
`  ( ( * `
 A )  x.  B ) )  e. 
_V
85, 6, 7ovmpt2a 6204 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 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988    x. cmul 8995   *ccj 11901   Imcim 11903
This theorem is referenced by:  sigarim  27817  sigarac  27818  sigaraf  27819  sigarmf  27820  sigarls  27823  sigarid  27824  sigardiv  27827  sharhght  27831
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086
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