Theorem sigaradd 41055

Description: Subtracting (double) area of A D C from A B C yields the (double) area of D B C. (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Hypotheses

Ref	Expression
sharhght.sigar	|- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
sharhght.a	|- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
sharhght.b	|- ( ph -> ( D e. CC /\ ( ( A - D ) G ( B - D ) ) = 0 ) )

Assertion

Ref	Expression
sigaradd	|- ( ph -> ( ( ( B - C ) G ( A - C ) ) - ( ( D - C ) G ( A - C ) ) ) = ( ( B - C ) G ( D - C ) ) )

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

Allowed substitution hints:   ph( x, y)   G( x, y)

Proof of Theorem sigaradd

Step	Hyp	Ref	Expression
1		sharhght.a	. . . . . . . 8 |- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
2	1	simp1d 1073	. . . . . . 7 |- ( ph -> A e. CC )
3	1	simp3d 1075	. . . . . . 7 |- ( ph -> C e. CC )
4		sharhght.b	. . . . . . . 8 |- ( ph -> ( D e. CC /\ ( ( A - D ) G ( B - D ) ) = 0 ) )
5	4	simpld 475	. . . . . . 7 |- ( ph -> D e. CC )
6	2, 3, 5	nnncan1d 10426	. . . . . 6 |- ( ph -> ( ( A - C ) - ( A - D ) ) = ( D - C ) )
7	6	oveq2d 6666	. . . . 5 |- ( ph -> ( ( B - D ) G ( ( A - C ) - ( A - D ) ) ) = ( ( B - D ) G ( D - C ) ) )
8	1	simp2d 1074	. . . . . . 7 |- ( ph -> B e. CC )
9	8, 5	subcld 10392	. . . . . 6 |- ( ph -> ( B - D ) e. CC )
10	2, 3	subcld 10392	. . . . . 6 |- ( ph -> ( A - C ) e. CC )
11	2, 5	subcld 10392	. . . . . 6 |- ( ph -> ( A - D ) e. CC )
12		sharhght.sigar	. . . . . . 7 |- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
13	12	sigarms 41045	. . . . . 6 |- ( ( ( B - D ) e. CC /\ ( A - C ) e. CC /\ ( A - D ) e. CC ) -> ( ( B - D ) G ( ( A - C ) - ( A - D ) ) ) = ( ( ( B - D ) G ( A - C ) ) - ( ( B - D ) G ( A - D ) ) ) )
14	9, 10, 11, 13	syl3anc 1326	. . . . 5 |- ( ph -> ( ( B - D ) G ( ( A - C ) - ( A - D ) ) ) = ( ( ( B - D ) G ( A - C ) ) - ( ( B - D ) G ( A - D ) ) ) )
15	7, 14	eqtr3d 2658	. . . 4 |- ( ph -> ( ( B - D ) G ( D - C ) ) = ( ( ( B - D ) G ( A - C ) ) - ( ( B - D ) G ( A - D ) ) ) )
16	12	sigarac 41041	. . . . . . . . 9 |- ( ( ( A - D ) e. CC /\ ( B - D ) e. CC ) -> ( ( A - D ) G ( B - D ) ) = -u ( ( B - D ) G ( A - D ) ) )
17	11, 9, 16	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( A - D ) G ( B - D ) ) = -u ( ( B - D ) G ( A - D ) ) )
18	4	simprd 479	. . . . . . . 8 |- ( ph -> ( ( A - D ) G ( B - D ) ) = 0 )
19	17, 18	eqtr3d 2658	. . . . . . 7 |- ( ph -> -u ( ( B - D ) G ( A - D ) ) = 0 )
20	19	negeqd 10275	. . . . . 6 |- ( ph -> -u -u ( ( B - D ) G ( A - D ) ) = -u 0 )
21	9, 11	jca 554	. . . . . . . 8 |- ( ph -> ( ( B - D ) e. CC /\ ( A - D ) e. CC ) )
22	12, 21	sigarimcd 41051	. . . . . . 7 |- ( ph -> ( ( B - D ) G ( A - D ) ) e. CC )
23	22	negnegd 10383	. . . . . 6 |- ( ph -> -u -u ( ( B - D ) G ( A - D ) ) = ( ( B - D ) G ( A - D ) ) )
24		neg0 10327	. . . . . . 7 |- -u 0 = 0
25	24	a1i 11	. . . . . 6 |- ( ph -> -u 0 = 0 )
26	20, 23, 25	3eqtr3d 2664	. . . . 5 |- ( ph -> ( ( B - D ) G ( A - D ) ) = 0 )
27	26	oveq2d 6666	. . . 4 |- ( ph -> ( ( ( B - D ) G ( A - C ) ) - ( ( B - D ) G ( A - D ) ) ) = ( ( ( B - D ) G ( A - C ) ) - 0 ) )
28	9, 10	jca 554	. . . . . 6 |- ( ph -> ( ( B - D ) e. CC /\ ( A - C ) e. CC ) )
29	12, 28	sigarimcd 41051	. . . . 5 |- ( ph -> ( ( B - D ) G ( A - C ) ) e. CC )
30	29	subid1d 10381	. . . 4 |- ( ph -> ( ( ( B - D ) G ( A - C ) ) - 0 ) = ( ( B - D ) G ( A - C ) ) )
31	15, 27, 30	3eqtrd 2660	. . 3 |- ( ph -> ( ( B - D ) G ( D - C ) ) = ( ( B - D ) G ( A - C ) ) )
32	8, 5, 3	nnncan2d 10427	. . . 4 |- ( ph -> ( ( B - C ) - ( D - C ) ) = ( B - D ) )
33	32	oveq1d 6665	. . 3 |- ( ph -> ( ( ( B - C ) - ( D - C ) ) G ( A - C ) ) = ( ( B - D ) G ( A - C ) ) )
34	8, 3	subcld 10392	. . . 4 |- ( ph -> ( B - C ) e. CC )
35	5, 3	subcld 10392	. . . 4 |- ( ph -> ( D - C ) e. CC )
36	12	sigarmf 41043	. . . 4 |- ( ( ( B - C ) e. CC /\ ( A - C ) e. CC /\ ( D - C ) e. CC ) -> ( ( ( B - C ) - ( D - C ) ) G ( A - C ) ) = ( ( ( B - C ) G ( A - C ) ) - ( ( D - C ) G ( A - C ) ) ) )
37	34, 10, 35, 36	syl3anc 1326	. . 3 |- ( ph -> ( ( ( B - C ) - ( D - C ) ) G ( A - C ) ) = ( ( ( B - C ) G ( A - C ) ) - ( ( D - C ) G ( A - C ) ) ) )
38	31, 33, 37	3eqtr2rd 2663	. 2 |- ( ph -> ( ( ( B - C ) G ( A - C ) ) - ( ( D - C ) G ( A - C ) ) ) = ( ( B - D ) G ( D - C ) ) )
39	3, 5	subcld 10392	. . . 4 |- ( ph -> ( C - D ) e. CC )
40		1red 10055	. . . . 5 |- ( ph -> 1 e. RR )
41	40	renegcld 10457	. . . 4 |- ( ph -> -u 1 e. RR )
42	12	sigarls 41046	. . . 4 |- ( ( ( B - D ) e. CC /\ ( C - D ) e. CC /\ -u 1 e. RR ) -> ( ( B - D ) G ( ( C - D ) x. -u 1 ) ) = ( ( ( B - D ) G ( C - D ) ) x. -u 1 ) )
43	9, 39, 41, 42	syl3anc 1326	. . 3 |- ( ph -> ( ( B - D ) G ( ( C - D ) x. -u 1 ) ) = ( ( ( B - D ) G ( C - D ) ) x. -u 1 ) )
44	39	mulm1d 10482	. . . . 5 |- ( ph -> ( -u 1 x. ( C - D ) ) = -u ( C - D ) )
45		1cnd 10056	. . . . . . 7 |- ( ph -> 1 e. CC )
46	45	negcld 10379	. . . . . 6 |- ( ph -> -u 1 e. CC )
47	46, 39	mulcomd 10061	. . . . 5 |- ( ph -> ( -u 1 x. ( C - D ) ) = ( ( C - D ) x. -u 1 ) )
48	3, 5	negsubdi2d 10408	. . . . 5 |- ( ph -> -u ( C - D ) = ( D - C ) )
49	44, 47, 48	3eqtr3d 2664	. . . 4 |- ( ph -> ( ( C - D ) x. -u 1 ) = ( D - C ) )
50	49	oveq2d 6666	. . 3 |- ( ph -> ( ( B - D ) G ( ( C - D ) x. -u 1 ) ) = ( ( B - D ) G ( D - C ) ) )
51	9, 39	jca 554	. . . . . 6 |- ( ph -> ( ( B - D ) e. CC /\ ( C - D ) e. CC ) )
52	12, 51	sigarimcd 41051	. . . . 5 |- ( ph -> ( ( B - D ) G ( C - D ) ) e. CC )
53	52	mulm1d 10482	. . . 4 |- ( ph -> ( -u 1 x. ( ( B - D ) G ( C - D ) ) ) = -u ( ( B - D ) G ( C - D ) ) )
54	52, 46	mulcomd 10061	. . . 4 |- ( ph -> ( ( ( B - D ) G ( C - D ) ) x. -u 1 ) = ( -u 1 x. ( ( B - D ) G ( C - D ) ) ) )
55	12	sigarac 41041	. . . . 5 |- ( ( ( C - D ) e. CC /\ ( B - D ) e. CC ) -> ( ( C - D ) G ( B - D ) ) = -u ( ( B - D ) G ( C - D ) ) )
56	39, 9, 55	syl2anc 693	. . . 4 |- ( ph -> ( ( C - D ) G ( B - D ) ) = -u ( ( B - D ) G ( C - D ) ) )
57	53, 54, 56	3eqtr4d 2666	. . 3 |- ( ph -> ( ( ( B - D ) G ( C - D ) ) x. -u 1 ) = ( ( C - D ) G ( B - D ) ) )
58	43, 50, 57	3eqtr3d 2664	. 2 |- ( ph -> ( ( B - D ) G ( D - C ) ) = ( ( C - D ) G ( B - D ) ) )
59	12	sigarperm 41049	. . 3 |- ( ( C e. CC /\ B e. CC /\ D e. CC ) -> ( ( C - D ) G ( B - D ) ) = ( ( B - C ) G ( D - C ) ) )
60	3, 8, 5, 59	syl3anc 1326	. 2 |- ( ph -> ( ( C - D ) G ( B - D ) ) = ( ( B - C ) G ( D - C ) ) )
61	38, 58, 60	3eqtrd 2660	1 |- ( ph -> ( ( ( B - C ) G ( A - C ) ) - ( ( D - C ) G ( A - C ) ) ) = ( ( B - C ) G ( D - 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   0cc0 9936   1c1 9937   x. cmul 9941   - cmin 10266   -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:  cevathlem2  41057