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Theorem sigarval 37859
 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
Assertion
Ref Expression
sigarval
Distinct variable groups:   ,,   ,,
Allowed substitution hints:   (,)

Proof of Theorem sigarval
StepHypRef Expression
1 simpl 458 . . . . 5
21fveq2d 5885 . . . 4
3 simpr 462 . . . 4
42, 3oveq12d 6323 . . 3
54fveq2d 5885 . 2
6 sigar . 2
7 fvex 5891 . 2
85, 6, 7ovmpt2a 6441 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1870  cfv 5601  (class class class)co 6305   cmpt2 6307  cc 9536   cmul 9543  ccj 13138  cim 13140 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310 This theorem is referenced by:  sigarim  37860  sigarac  37861  sigaraf  37862  sigarmf  37863  sigarls  37866  sigarid  37867  sigardiv  37870  sharhght  37874
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