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Theorem sigarval 27942
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 443 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  x  =  A )
21fveq2d 5545 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  ( * `  x
)  =  ( * `
 A ) )
3 simpr 447 . . . 4  |-  ( ( x  =  A  /\  y  =  B )  ->  y  =  B )
42, 3oveq12d 5892 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ( * `  x )  x.  y
)  =  ( ( * `  A )  x.  B ) )
54fveq2d 5545 . 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 5555 . 2  |-  ( Im
`  ( ( * `
 A )  x.  B ) )  e. 
_V
85, 6, 7ovmpt2a 5994 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 358    = wceq 1632    e. wcel 1696   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   CCcc 8751    x. cmul 8758   *ccj 11597   Imcim 11599
This theorem is referenced by:  sigarim  27943  sigarac  27944  sigaraf  27945  sigarmf  27946  sigarls  27949  sigarid  27950  sigardiv  27953  sharhght  27957
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879
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