Theorem sigaras 41044

Description: Signed area is additive by the second argument. (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
sigaras	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A G ( B + C ) ) = ( ( A G B ) + ( A G C ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

Allowed substitution hints:   G( x, y)

Proof of Theorem sigaras

Step	Hyp	Ref	Expression
1		simp1 1061	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> A e. CC )
2		simp2 1062	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
3		simp3 1063	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
4	2, 3	addcld 10059	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B + C ) e. CC )
5		sigar	. . . 4 |- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
6	5	sigarac 41041	. . 3 |- ( ( A e. CC /\ ( B + C ) e. CC ) -> ( A G ( B + C ) ) = -u ( ( B + C ) G A ) )
7	1, 4, 6	syl2anc 693	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A G ( B + C ) ) = -u ( ( B + C ) G A ) )
8	5	sigaraf 41042	. . . . 5 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> ( ( B + C ) G A ) = ( ( B G A ) + ( C G A ) ) )
9	8	negeqd 10275	. . . 4 |- ( ( B e. CC /\ A e. CC /\ C e. CC ) -> -u ( ( B + C ) G A ) = -u ( ( B G A ) + ( C G A ) ) )
10	9	3com12 1269	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> -u ( ( B + C ) G A ) = -u ( ( B G A ) + ( C G A ) ) )
11		3simpa 1058	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A e. CC /\ B e. CC ) )
12	11	ancomd 467	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B e. CC /\ A e. CC ) )
13	5	sigarim 41040	. . . . . 6 |- ( ( B e. CC /\ A e. CC ) -> ( B G A ) e. RR )
14	12, 13	syl 17	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B G A ) e. RR )
15	14	recnd 10068	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( B G A ) e. CC )
16		3simpb 1059	. . . . . . 7 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A e. CC /\ C e. CC ) )
17	16	ancomd 467	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C e. CC /\ A e. CC ) )
18	5	sigarim 41040	. . . . . 6 |- ( ( C e. CC /\ A e. CC ) -> ( C G A ) e. RR )
19	17, 18	syl 17	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C G A ) e. RR )
20	19	recnd 10068	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C G A ) e. CC )
21	15, 20	negdid 10405	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> -u ( ( B G A ) + ( C G A ) ) = ( -u ( B G A ) + -u ( C G A ) ) )
22	10, 21	eqtrd 2656	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> -u ( ( B + C ) G A ) = ( -u ( B G A ) + -u ( C G A ) ) )
23	5	sigarac 41041	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A G B ) = -u ( B G A ) )
24	1, 2, 23	syl2anc 693	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A G B ) = -u ( B G A ) )
25	24	eqcomd 2628	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> -u ( B G A ) = ( A G B ) )
26	5	sigarac 41041	. . . . 5 |- ( ( A e. CC /\ C e. CC ) -> ( A G C ) = -u ( C G A ) )
27	1, 3, 26	syl2anc 693	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A G C ) = -u ( C G A ) )
28	27	eqcomd 2628	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> -u ( C G A ) = ( A G C ) )
29	25, 28	oveq12d 6668	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( -u ( B G A ) + -u ( C G A ) ) = ( ( A G B ) + ( A G C ) ) )
30	7, 22, 29	3eqtrd 2660	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A G ( B + C ) ) = ( ( A G B ) + ( A G C ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   + caddc 9939   x. cmul 9941   -ucneg 10267   *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-8 1992  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-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-2 11079  df-cj 13839  df-re 13840  df-im 13841

This theorem is referenced by: (None)