Theorem cevathlem2 41057

Description: Ceva's theorem second lemma. Relate (doubled) areas of triangles C A O and A B O with of segments B D and D C. (Contributed by Saveliy Skresanov, 24-Sep-2017.)

Hypotheses

Ref	Expression
cevath.sigar	|- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
cevath.a	|- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
cevath.b	|- ( ph -> ( F e. CC /\ D e. CC /\ E e. CC ) )
cevath.c	|- ( ph -> O e. CC )
cevath.d	|- ( ph -> ( ( ( A - O ) G ( D - O ) ) = 0 /\ ( ( B - O ) G ( E - O ) ) = 0 /\ ( ( C - O ) G ( F - O ) ) = 0 ) )
cevath.e	|- ( ph -> ( ( ( A - F ) G ( B - F ) ) = 0 /\ ( ( B - D ) G ( C - D ) ) = 0 /\ ( ( C - E ) G ( A - E ) ) = 0 ) )
cevath.f	|- ( ph -> ( ( ( A - O ) G ( B - O ) ) =/= 0 /\ ( ( B - O ) G ( C - O ) ) =/= 0 /\ ( ( C - O ) G ( A - O ) ) =/= 0 ) )

Assertion

Ref	Expression
cevathlem2	|- ( ph -> ( ( ( C - O ) G ( A - O ) ) x. ( B - D ) ) = ( ( ( A - O ) G ( B - O ) ) x. ( D - C ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y   x, O, y   x, E, y   x, F, y

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

Proof of Theorem cevathlem2

Step	Hyp	Ref	Expression
1		cevath.sigar	. . . . . . 7 |- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
2		cevath.b	. . . . . . . . 9 |- ( ph -> ( F e. CC /\ D e. CC /\ E e. CC ) )
3	2	simp2d 1074	. . . . . . . 8 |- ( ph -> D e. CC )
4		cevath.a	. . . . . . . . 9 |- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
5	4	simp1d 1073	. . . . . . . 8 |- ( ph -> A e. CC )
6	4	simp2d 1074	. . . . . . . 8 |- ( ph -> B e. CC )
7	3, 5, 6	3jca 1242	. . . . . . 7 |- ( ph -> ( D e. CC /\ A e. CC /\ B e. CC ) )
8		cevath.c	. . . . . . . 8 |- ( ph -> O e. CC )
9	5, 8	subcld 10392	. . . . . . . . . 10 |- ( ph -> ( A - O ) e. CC )
10	3, 8	subcld 10392	. . . . . . . . . 10 |- ( ph -> ( D - O ) e. CC )
11	9, 10	jca 554	. . . . . . . . 9 |- ( ph -> ( ( A - O ) e. CC /\ ( D - O ) e. CC ) )
12		cevath.d	. . . . . . . . . 10 |- ( ph -> ( ( ( A - O ) G ( D - O ) ) = 0 /\ ( ( B - O ) G ( E - O ) ) = 0 /\ ( ( C - O ) G ( F - O ) ) = 0 ) )
13	12	simp1d 1073	. . . . . . . . 9 |- ( ph -> ( ( A - O ) G ( D - O ) ) = 0 )
14	1, 11, 13	sigariz 41052	. . . . . . . 8 |- ( ph -> ( ( D - O ) G ( A - O ) ) = 0 )
15	8, 14	jca 554	. . . . . . 7 |- ( ph -> ( O e. CC /\ ( ( D - O ) G ( A - O ) ) = 0 ) )
16	1, 7, 15	sigaradd 41055	. . . . . 6 |- ( ph -> ( ( ( A - B ) G ( D - B ) ) - ( ( O - B ) G ( D - B ) ) ) = ( ( A - B ) G ( O - B ) ) )
17	1	sigarperm 41049	. . . . . . 7 |- ( ( B e. CC /\ A e. CC /\ O e. CC ) -> ( ( B - O ) G ( A - O ) ) = ( ( A - B ) G ( O - B ) ) )
18	6, 5, 8, 17	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( B - O ) G ( A - O ) ) = ( ( A - B ) G ( O - B ) ) )
19	16, 18	eqtr4d 2659	. . . . 5 |- ( ph -> ( ( ( A - B ) G ( D - B ) ) - ( ( O - B ) G ( D - B ) ) ) = ( ( B - O ) G ( A - O ) ) )
20	19	oveq1d 6665	. . . 4 |- ( ph -> ( ( ( ( A - B ) G ( D - B ) ) - ( ( O - B ) G ( D - B ) ) ) x. ( C - D ) ) = ( ( ( B - O ) G ( A - O ) ) x. ( C - D ) ) )
21	5, 6	subcld 10392	. . . . . . 7 |- ( ph -> ( A - B ) e. CC )
22	3, 6	subcld 10392	. . . . . . 7 |- ( ph -> ( D - B ) e. CC )
23	21, 22	jca 554	. . . . . 6 |- ( ph -> ( ( A - B ) e. CC /\ ( D - B ) e. CC ) )
24	1, 23	sigarimcd 41051	. . . . 5 |- ( ph -> ( ( A - B ) G ( D - B ) ) e. CC )
25	8, 6	subcld 10392	. . . . . . 7 |- ( ph -> ( O - B ) e. CC )
26	25, 22	jca 554	. . . . . 6 |- ( ph -> ( ( O - B ) e. CC /\ ( D - B ) e. CC ) )
27	1, 26	sigarimcd 41051	. . . . 5 |- ( ph -> ( ( O - B ) G ( D - B ) ) e. CC )
28	4	simp3d 1075	. . . . . 6 |- ( ph -> C e. CC )
29	28, 3	subcld 10392	. . . . 5 |- ( ph -> ( C - D ) e. CC )
30	24, 27, 29	subdird 10487	. . . 4 |- ( ph -> ( ( ( ( A - B ) G ( D - B ) ) - ( ( O - B ) G ( D - B ) ) ) x. ( C - D ) ) = ( ( ( ( A - B ) G ( D - B ) ) x. ( C - D ) ) - ( ( ( O - B ) G ( D - B ) ) x. ( C - D ) ) ) )
31	20, 30	eqtr3d 2658	. . 3 |- ( ph -> ( ( ( B - O ) G ( A - O ) ) x. ( C - D ) ) = ( ( ( ( A - B ) G ( D - B ) ) x. ( C - D ) ) - ( ( ( O - B ) G ( D - B ) ) x. ( C - D ) ) ) )
32	6, 28, 5	3jca 1242	. . . . 5 |- ( ph -> ( B e. CC /\ C e. CC /\ A e. CC ) )
33		cevath.e	. . . . . . 7 |- ( ph -> ( ( ( A - F ) G ( B - F ) ) = 0 /\ ( ( B - D ) G ( C - D ) ) = 0 /\ ( ( C - E ) G ( A - E ) ) = 0 ) )
34	33	simp2d 1074	. . . . . 6 |- ( ph -> ( ( B - D ) G ( C - D ) ) = 0 )
35	3, 34	jca 554	. . . . 5 |- ( ph -> ( D e. CC /\ ( ( B - D ) G ( C - D ) ) = 0 ) )
36	1, 32, 35	sharhght 41054	. . . 4 |- ( ph -> ( ( ( A - B ) G ( D - B ) ) x. ( C - D ) ) = ( ( ( A - C ) G ( D - C ) ) x. ( B - D ) ) )
37	6, 28, 8	3jca 1242	. . . . 5 |- ( ph -> ( B e. CC /\ C e. CC /\ O e. CC ) )
38	1, 37, 35	sharhght 41054	. . . 4 |- ( ph -> ( ( ( O - B ) G ( D - B ) ) x. ( C - D ) ) = ( ( ( O - C ) G ( D - C ) ) x. ( B - D ) ) )
39	36, 38	oveq12d 6668	. . 3 |- ( ph -> ( ( ( ( A - B ) G ( D - B ) ) x. ( C - D ) ) - ( ( ( O - B ) G ( D - B ) ) x. ( C - D ) ) ) = ( ( ( ( A - C ) G ( D - C ) ) x. ( B - D ) ) - ( ( ( O - C ) G ( D - C ) ) x. ( B - D ) ) ) )
40	5, 28	subcld 10392	. . . . . . 7 |- ( ph -> ( A - C ) e. CC )
41	3, 28	subcld 10392	. . . . . . 7 |- ( ph -> ( D - C ) e. CC )
42	1	sigarim 41040	. . . . . . 7 |- ( ( ( A - C ) e. CC /\ ( D - C ) e. CC ) -> ( ( A - C ) G ( D - C ) ) e. RR )
43	40, 41, 42	syl2anc 693	. . . . . 6 |- ( ph -> ( ( A - C ) G ( D - C ) ) e. RR )
44	43	recnd 10068	. . . . 5 |- ( ph -> ( ( A - C ) G ( D - C ) ) e. CC )
45	8, 28	subcld 10392	. . . . . . 7 |- ( ph -> ( O - C ) e. CC )
46	45, 41	jca 554	. . . . . 6 |- ( ph -> ( ( O - C ) e. CC /\ ( D - C ) e. CC ) )
47	1, 46	sigarimcd 41051	. . . . 5 |- ( ph -> ( ( O - C ) G ( D - C ) ) e. CC )
48	6, 3	subcld 10392	. . . . 5 |- ( ph -> ( B - D ) e. CC )
49	44, 47, 48	subdird 10487	. . . 4 |- ( ph -> ( ( ( ( A - C ) G ( D - C ) ) - ( ( O - C ) G ( D - C ) ) ) x. ( B - D ) ) = ( ( ( ( A - C ) G ( D - C ) ) x. ( B - D ) ) - ( ( ( O - C ) G ( D - C ) ) x. ( B - D ) ) ) )
50	3, 5, 28	3jca 1242	. . . . . . 7 |- ( ph -> ( D e. CC /\ A e. CC /\ C e. CC ) )
51	1, 50, 15	sigaradd 41055	. . . . . 6 |- ( ph -> ( ( ( A - C ) G ( D - C ) ) - ( ( O - C ) G ( D - C ) ) ) = ( ( A - C ) G ( O - C ) ) )
52	1	sigarperm 41049	. . . . . . 7 |- ( ( C e. CC /\ A e. CC /\ O e. CC ) -> ( ( C - O ) G ( A - O ) ) = ( ( A - C ) G ( O - C ) ) )
53	28, 5, 8, 52	syl3anc 1326	. . . . . 6 |- ( ph -> ( ( C - O ) G ( A - O ) ) = ( ( A - C ) G ( O - C ) ) )
54	51, 53	eqtr4d 2659	. . . . 5 |- ( ph -> ( ( ( A - C ) G ( D - C ) ) - ( ( O - C ) G ( D - C ) ) ) = ( ( C - O ) G ( A - O ) ) )
55	54	oveq1d 6665	. . . 4 |- ( ph -> ( ( ( ( A - C ) G ( D - C ) ) - ( ( O - C ) G ( D - C ) ) ) x. ( B - D ) ) = ( ( ( C - O ) G ( A - O ) ) x. ( B - D ) ) )
56	49, 55	eqtr3d 2658	. . 3 |- ( ph -> ( ( ( ( A - C ) G ( D - C ) ) x. ( B - D ) ) - ( ( ( O - C ) G ( D - C ) ) x. ( B - D ) ) ) = ( ( ( C - O ) G ( A - O ) ) x. ( B - D ) ) )
57	31, 39, 56	3eqtrrd 2661	. 2 |- ( ph -> ( ( ( C - O ) G ( A - O ) ) x. ( B - D ) ) = ( ( ( B - O ) G ( A - O ) ) x. ( C - D ) ) )
58	6, 8	subcld 10392	. . . 4 |- ( ph -> ( B - O ) e. CC )
59	1	sigarac 41041	. . . 4 |- ( ( ( B - O ) e. CC /\ ( A - O ) e. CC ) -> ( ( B - O ) G ( A - O ) ) = -u ( ( A - O ) G ( B - O ) ) )
60	58, 9, 59	syl2anc 693	. . 3 |- ( ph -> ( ( B - O ) G ( A - O ) ) = -u ( ( A - O ) G ( B - O ) ) )
61	60	oveq1d 6665	. 2 |- ( ph -> ( ( ( B - O ) G ( A - O ) ) x. ( C - D ) ) = ( -u ( ( A - O ) G ( B - O ) ) x. ( C - D ) ) )
62	9, 58	jca 554	. . . . 5 |- ( ph -> ( ( A - O ) e. CC /\ ( B - O ) e. CC ) )
63	1, 62	sigarimcd 41051	. . . 4 |- ( ph -> ( ( A - O ) G ( B - O ) ) e. CC )
64		mulneg12 10468	. . . 4 |- ( ( ( ( A - O ) G ( B - O ) ) e. CC /\ ( C - D ) e. CC ) -> ( -u ( ( A - O ) G ( B - O ) ) x. ( C - D ) ) = ( ( ( A - O ) G ( B - O ) ) x. -u ( C - D ) ) )
65	63, 29, 64	syl2anc 693	. . 3 |- ( ph -> ( -u ( ( A - O ) G ( B - O ) ) x. ( C - D ) ) = ( ( ( A - O ) G ( B - O ) ) x. -u ( C - D ) ) )
66	28, 3	negsubdi2d 10408	. . . 4 |- ( ph -> -u ( C - D ) = ( D - C ) )
67	66	oveq2d 6666	. . 3 |- ( ph -> ( ( ( A - O ) G ( B - O ) ) x. -u ( C - D ) ) = ( ( ( A - O ) G ( B - O ) ) x. ( D - C ) ) )
68	65, 67	eqtrd 2656	. 2 |- ( ph -> ( -u ( ( A - O ) G ( B - O ) ) x. ( C - D ) ) = ( ( ( A - O ) G ( B - O ) ) x. ( D - C ) ) )
69	57, 61, 68	3eqtrd 2660	1 |- ( ph -> ( ( ( C - O ) G ( A - O ) ) x. ( B - D ) ) = ( ( ( A - O ) G ( B - O ) ) x. ( D - C ) ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   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:  cevath  41058