Theorem sigarcol 29874

Description: Given three points A, B and C such that -. A = B, the point C lies on the line going through A and B iff the corresponding signed area is zero. That justifies the usage of signed area as a collinearity indicator. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Hypotheses

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

Assertion

Ref	Expression
sigarcol	|- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 <-> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) ) )

Distinct variable groups:   x, t, y, A   t, B, x, y   t, C, x, y   t, G   ph, t

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

Proof of Theorem sigarcol

Step	Hyp	Ref	Expression
1		sigarcol.sigar	. . . . 5 |- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
2		sigarcol.a	. . . . . . . 8 |- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
3	2	simp2d 1001	. . . . . . 7 |- ( ph -> B e. CC )
4	2	simp3d 1002	. . . . . . 7 |- ( ph -> C e. CC )
5	2	simp1d 1000	. . . . . . 7 |- ( ph -> A e. CC )
6	3, 4, 5	3jca 1168	. . . . . 6 |- ( ph -> ( B e. CC /\ C e. CC /\ A e. CC ) )
7	6	adantr 465	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B e. CC /\ C e. CC /\ A e. CC ) )
8		sigarcol.b	. . . . . 6 |- ( ph -> -. A = B )
9	8	adantr 465	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> -. A = B )
10	1	sigarperm 29870	. . . . . . . . 9 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) G ( B - C ) ) = ( ( B - A ) G ( C - A ) ) )
11	2, 10	syl 16	. . . . . . . 8 |- ( ph -> ( ( A - C ) G ( B - C ) ) = ( ( B - A ) G ( C - A ) ) )
12	1	sigarperm 29870	. . . . . . . . 9 |- ( ( B e. CC /\ C e. CC /\ A e. CC ) -> ( ( B - A ) G ( C - A ) ) = ( ( C - B ) G ( A - B ) ) )
13	6, 12	syl 16	. . . . . . . 8 |- ( ph -> ( ( B - A ) G ( C - A ) ) = ( ( C - B ) G ( A - B ) ) )
14	11, 13	eqtrd 2470	. . . . . . 7 |- ( ph -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
15	14	eqeq1d 2446	. . . . . 6 |- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 <-> ( ( C - B ) G ( A - B ) ) = 0 ) )
16	15	biimpa 484	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) G ( A - B ) ) = 0 )
17	1, 7, 9, 16	sigardiv 29871	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) / ( A - B ) ) e. RR )
18	4, 3	subcld 9711	. . . . . . . 8 |- ( ph -> ( C - B ) e. CC )
19	18	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( C - B ) e. CC )
20	5, 3	subcld 9711	. . . . . . . 8 |- ( ph -> ( A - B ) e. CC )
21	20	adantr 465	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) e. CC )
22	5	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A e. CC )
23	3	adantr 465	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> B e. CC )
24	9	neneqad 2676	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A =/= B )
25	22, 23, 24	subne0d 9720	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) =/= 0 )
26	19, 21, 25	divcan1d 10100	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) = ( C - B ) )
27	26	oveq2d 6102	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) = ( B + ( C - B ) ) )
28	4	adantr 465	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C e. CC )
29	23, 28	pncan3d 9714	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( C - B ) ) = C )
30	27, 29	eqtr2d 2471	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
31		oveq1 6093	. . . . . . 7 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( t x. ( A - B ) ) = ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) )
32	31	oveq2d 6102	. . . . . 6 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( B + ( t x. ( A - B ) ) ) = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
33	32	eqeq2d 2449	. . . . 5 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( C = ( B + ( t x. ( A - B ) ) ) <-> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) ) )
34	33	rspcev 3068	. . . 4 |- ( ( ( ( C - B ) / ( A - B ) ) e. RR /\ C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) ) -> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) )
35	17, 30, 34	syl2anc 661	. . 3 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) )
36	35	ex 434	. 2 |- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 -> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) ) )
37	14	3ad2ant1 1009	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
38		simp3 990	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> C = ( B + ( t x. ( A - B ) ) ) )
39	38	oveq1d 6101	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( ( B + ( t x. ( A - B ) ) ) - B ) )
40	3	3ad2ant1 1009	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> B e. CC )
41		simp2 989	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. RR )
42	41	recnd 9404	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. CC )
43	5	3ad2ant1 1009	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> A e. CC )
44	43, 40	subcld 9711	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( A - B ) e. CC )
45	42, 44	mulcld 9398	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) e. CC )
46	40, 45	pncan2d 9713	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( B + ( t x. ( A - B ) ) ) - B ) = ( t x. ( A - B ) ) )
47	39, 46	eqtrd 2470	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( t x. ( A - B ) ) )
48	47	oveq1d 6101	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( C - B ) G ( A - B ) ) = ( ( t x. ( A - B ) ) G ( A - B ) ) )
49	42, 44	mulcomd 9399	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) = ( ( A - B ) x. t ) )
50	49	oveq1d 6101	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( t x. ( A - B ) ) G ( A - B ) ) = ( ( ( A - B ) x. t ) G ( A - B ) ) )
51	48, 50	eqtrd 2470	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( C - B ) G ( A - B ) ) = ( ( ( A - B ) x. t ) G ( A - B ) ) )
52	44, 42	mulcld 9398	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) x. t ) e. CC )
53	1	sigarac 29862	. . . . . 6 |- ( ( ( ( A - B ) x. t ) e. CC /\ ( A - B ) e. CC ) -> ( ( ( A - B ) x. t ) G ( A - B ) ) = -u ( ( A - B ) G ( ( A - B ) x. t ) ) )
54	52, 44, 53	syl2anc 661	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) x. t ) G ( A - B ) ) = -u ( ( A - B ) G ( ( A - B ) x. t ) ) )
55	1	sigarls 29867	. . . . . . . 8 |- ( ( ( A - B ) e. CC /\ ( A - B ) e. CC /\ t e. RR ) -> ( ( A - B ) G ( ( A - B ) x. t ) ) = ( ( ( A - B ) G ( A - B ) ) x. t ) )
56	44, 44, 41, 55	syl3anc 1218	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( ( A - B ) x. t ) ) = ( ( ( A - B ) G ( A - B ) ) x. t ) )
57	1	sigarid 29868	. . . . . . . . 9 |- ( ( A - B ) e. CC -> ( ( A - B ) G ( A - B ) ) = 0 )
58	44, 57	syl 16	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( A - B ) ) = 0 )
59	58	oveq1d 6101	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) G ( A - B ) ) x. t ) = ( 0 x. t ) )
60	42	mul02d 9559	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( 0 x. t ) = 0 )
61	56, 59, 60	3eqtrd 2474	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( ( A - B ) x. t ) ) = 0 )
62	61	negeqd 9596	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> -u ( ( A - B ) G ( ( A - B ) x. t ) ) = -u 0 )
63		neg0 9647	. . . . . 6 |- -u 0 = 0
64	63	a1i 11	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> -u 0 = 0 )
65	54, 62, 64	3eqtrd 2474	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) x. t ) G ( A - B ) ) = 0 )
66	37, 51, 65	3eqtrd 2474	. . 3 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 )
67	66	rexlimdv3a 2838	. 2 |- ( ph -> ( E. t e. RR C = ( B + ( t x. ( A - B ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 ) )
68	36, 67	impbid 191	1 |- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 <-> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   E.wrex 2711   ` cfv 5413  (class class class)co 6086   e. cmpt2 6088   CCcc 9272   RRcr 9273   0cc0 9274   + caddc 9277   x. cmul 9279   - cmin 9587   -ucneg 9588   / cdiv 9985   *ccj 12577   Imcim 12579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-2 10372  df-cj 12580  df-re 12581  df-im 12582

This theorem is referenced by: (None)