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Theorem sigarval 27501
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 5665 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( * `  x
)  =  ( * `
 A ) )
3 simpr 448 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
42, 3oveq12d 6031 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( * `  x )  x.  y
)  =  ( ( * `  A )  x.  B ) )
54fveq2d 5665 . 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 5675 . 2  |-  ( Im
`  ( ( * `
 A )  x.  B ) )  e. 
_V
85, 6, 7ovmpt2a 6136 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 1717   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   CCcc 8914    x. cmul 8921   *ccj 11821   Imcim 11823
This theorem is referenced by:  sigarim  27502  sigarac  27503  sigaraf  27504  sigarmf  27505  sigarls  27508  sigarid  27509  sigardiv  27512  sharhght  27516
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-sbc 3098  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-iota 5351  df-fun 5389  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018
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