Theorem sigarcol 32035

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 1010	. . . . . . 7 |- ( ph -> B e. CC )
4	2	simp3d 1011	. . . . . . 7 |- ( ph -> C e. CC )
5	2	simp1d 1009	. . . . . . 7 |- ( ph -> A e. CC )
6	3, 4, 5	3jca 1177	. . . . . 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 32031	. . . . . . . . 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 32031	. . . . . . . . 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 2484	. . . . . . 7 |- ( ph -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
15	14	eqeq1d 2445	. . . . . 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 32032	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) / ( A - B ) ) e. RR )
18	4, 3	subcld 9936	. . . . . . . 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 9936	. . . . . . . 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	neqned 2646	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A =/= B )
25	22, 23, 24	subne0d 9945	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) =/= 0 )
26	19, 21, 25	divcan1d 10328	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) = ( C - B ) )
27	26	oveq2d 6297	. . . . 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 9939	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( C - B ) ) = C )
30	27, 29	eqtr2d 2485	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
31		oveq1 6288	. . . . . . 7 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( t x. ( A - B ) ) = ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) )
32	31	oveq2d 6297	. . . . . 6 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( B + ( t x. ( A - B ) ) ) = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
33	32	eqeq2d 2457	. . . . 5 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( C = ( B + ( t x. ( A - B ) ) ) <-> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) ) )
34	33	rspcev 3196	. . . 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 1018	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
38		simp3 999	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> C = ( B + ( t x. ( A - B ) ) ) )
39	38	oveq1d 6296	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( ( B + ( t x. ( A - B ) ) ) - B ) )
40	3	3ad2ant1 1018	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> B e. CC )
41		simp2 998	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. RR )
42	41	recnd 9625	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. CC )
43	5	3ad2ant1 1018	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> A e. CC )
44	43, 40	subcld 9936	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( A - B ) e. CC )
45	42, 44	mulcld 9619	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) e. CC )
46	40, 45	pncan2d 9938	. . . . . . 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 2484	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( t x. ( A - B ) ) )
48	47	oveq1d 6296	. . . . 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 9620	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) = ( ( A - B ) x. t ) )
50	49	oveq1d 6296	. . . . 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 2484	. . . 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 9619	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) x. t ) e. CC )
53	1	sigarac 32023	. . . . . 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 32028	. . . . . . . 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 1229	. . . . . . 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 32029	. . . . . . . . 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 6296	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) G ( A - B ) ) x. t ) = ( 0 x. t ) )
60	42	mul02d 9781	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( 0 x. t ) = 0 )
61	56, 59, 60	3eqtrd 2488	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( ( A - B ) x. t ) ) = 0 )
62	61	negeqd 9819	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> -u ( ( A - B ) G ( ( A - B ) x. t ) ) = -u 0 )
63		neg0 9870	. . . . . 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 2488	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) x. t ) G ( A - B ) ) = 0 )
66	37, 51, 65	3eqtrd 2488	. . 3 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 )
67	66	rexlimdv3a 2937	. 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 974   = wceq 1383   e. wcel 1804   E.wrex 2794   ` cfv 5578  (class class class)co 6281   |-> cmpt2 6283   CCcc 9493   RRcr 9494   0cc0 9495   + caddc 9498   x. cmul 9500   - cmin 9810   -ucneg 9811   / cdiv 10213   *ccj 12911   Imcim 12913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-2 10601  df-cj 12914  df-re 12915  df-im 12916

This theorem is referenced by: (None)