Theorem sigarval 41039

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

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( x = A /\ y = B ) -> x = A )
2	1	fveq2d 6195	. . . 4 |- ( ( x = A /\ y = B ) -> ( * ` x ) = ( * ` A ) )
3		simpr 477	. . . 4 |- ( ( x = A /\ y = B ) -> y = B )
4	2, 3	oveq12d 6668	. . 3 |- ( ( x = A /\ y = B ) -> ( ( * ` x ) x. y ) = ( ( * ` A ) x. B ) )
5	4	fveq2d 6195	. 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 6201	. 2 |- ( Im ` ( ( * ` A ) x. B ) ) e. _V
8	5, 6, 7	ovmpt2a 6791	1 |- ( ( A e. CC /\ B e. CC ) -> ( A G B ) = ( Im ` ( ( * ` A ) x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   x. cmul 9941   *ccj 13836   Imcim 13838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  sigarim  41040  sigarac  41041  sigaraf  41042  sigarmf  41043  sigarls  41046  sigarid  41047  sigardiv  41050  sharhght  41054