sigarperm - Mathbox for Saveliy Skresanov

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Theorem sigarperm 41049

Description: Signed area

acts as a double area of a triangle

. Here we prove that cyclically permuting the vertices doesn't change the area. (Contributed by Saveliy Skresanov, 20-Sep-2017.)

**Hypothesis**
Ref	Expression
sigar

**Assertion**
Ref	Expression
sigarperm	$/\$ $/\$

(

)

**Proof of Theorem sigarperm**
Step	Hyp	Ref	Expression
1		simp2 1062	. . . . . 6 $/\$ $/\$
2		simp3 1063	. . . . . 6 $/\$ $/\$
3		sigar	. . . . . . . 8
4	3	sigarim 41040	. . . . . . 7 $/\$
5	4	recnd 10068	. . . . . 6 $/\$
6	1, 2, 5	syl2anc 693	. . . . 5 $/\$ $/\$
7		simp1 1061	. . . . . 6 $/\$ $/\$
8	3	sigarim 41040	. . . . . . 7 $/\$
9	8	recnd 10068	. . . . . 6 $/\$
10	1, 7, 9	syl2anc 693	. . . . 5 $/\$ $/\$
11	6, 10	negsubd 10398	. . . 4 $/\$ $/\$
12	3	sigarac 41041	. . . . . . 7 $/\$
13	7, 1, 12	syl2anc 693	. . . . . 6 $/\$ $/\$
14	13	eqcomd 2628	. . . . 5 $/\$ $/\$
15	14	oveq2d 6666	. . . 4 $/\$ $/\$
16	11, 15	eqtr3d 2658	. . 3 $/\$ $/\$
17	16	oveq1d 6665	. 2 $/\$ $/\$
18	3	sigarexp 41048	. . 3 $/\$ $/\$
19	18	3comr 1273	. 2 $/\$ $/\$
20	3	sigarexp 41048	. . 3 $/\$ $/\$
21	3	sigarim 41040	. . . . . 6 $/\$
22	7, 1, 21	syl2anc 693	. . . . 5 $/\$ $/\$
23	22	recnd 10068	. . . 4 $/\$ $/\$
24	3	sigarim 41040	. . . . . 6 $/\$
25	7, 2, 24	syl2anc 693	. . . . 5 $/\$ $/\$
26	25	recnd 10068	. . . 4 $/\$ $/\$
27	3	sigarim 41040	. . . . . 6 $/\$
28	2, 1, 27	syl2anc 693	. . . . 5 $/\$ $/\$
29	28	recnd 10068	. . . 4 $/\$ $/\$
30	23, 26, 29	sub32d 10424	. . 3 $/\$ $/\$
31	6, 23	addcomd 10238	. . . . 5 $/\$ $/\$
32	3	sigarac 41041	. . . . . . . 8 $/\$
33	1, 2, 32	syl2anc 693	. . . . . . 7 $/\$ $/\$
34	33	eqcomd 2628	. . . . . 6 $/\$ $/\$
35	34	oveq2d 6666	. . . . 5 $/\$ $/\$
36	23, 29	negsubd 10398	. . . . 5 $/\$ $/\$
37	31, 35, 36	3eqtr2rd 2663	. . . 4 $/\$ $/\$
38	37	oveq1d 6665	. . 3 $/\$ $/\$
39	20, 30, 38	3eqtrd 2660	. 2 $/\$ $/\$
40	17, 19, 39	3eqtr4rd 2667	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

cfv 5888 (class class class)co 6650

cmpt2 6652

cc 9934

cr 9935

caddc 9939

cmul 9941

cmin 10266

cneg 10267

ccj 13836

cim 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: sigarcol 41053 sharhght 41054 sigaradd 41055 cevathlem2 41057