Theorem sigarcol 31548

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 1009	. . . . . . 7 |- ( ph -> B e. CC )
4	2	simp3d 1010	. . . . . . 7 |- ( ph -> C e. CC )
5	2	simp1d 1008	. . . . . . 7 |- ( ph -> A e. CC )
6	3, 4, 5	3jca 1176	. . . . . 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 31544	. . . . . . . . 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 31544	. . . . . . . . 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 2508	. . . . . . 7 |- ( ph -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
15	14	eqeq1d 2469	. . . . . 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 31545	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) / ( A - B ) ) e. RR )
18	4, 3	subcld 9926	. . . . . . . 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 9926	. . . . . . . 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 2670	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A =/= B )
25	22, 23, 24	subne0d 9935	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) =/= 0 )
26	19, 21, 25	divcan1d 10317	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) = ( C - B ) )
27	26	oveq2d 6298	. . . . 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 9929	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( C - B ) ) = C )
30	27, 29	eqtr2d 2509	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
31		oveq1 6289	. . . . . . 7 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( t x. ( A - B ) ) = ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) )
32	31	oveq2d 6298	. . . . . 6 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( B + ( t x. ( A - B ) ) ) = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
33	32	eqeq2d 2481	. . . . 5 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( C = ( B + ( t x. ( A - B ) ) ) <-> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) ) )
34	33	rspcev 3214	. . . 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 1017	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
38		simp3 998	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> C = ( B + ( t x. ( A - B ) ) ) )
39	38	oveq1d 6297	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( ( B + ( t x. ( A - B ) ) ) - B ) )
40	3	3ad2ant1 1017	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> B e. CC )
41		simp2 997	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. RR )
42	41	recnd 9618	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. CC )
43	5	3ad2ant1 1017	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> A e. CC )
44	43, 40	subcld 9926	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( A - B ) e. CC )
45	42, 44	mulcld 9612	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) e. CC )
46	40, 45	pncan2d 9928	. . . . . . 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 2508	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( t x. ( A - B ) ) )
48	47	oveq1d 6297	. . . . 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 9613	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) = ( ( A - B ) x. t ) )
50	49	oveq1d 6297	. . . . 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 2508	. . . 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 9612	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) x. t ) e. CC )
53	1	sigarac 31536	. . . . . 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 31541	. . . . . . . 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 1228	. . . . . . 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 31542	. . . . . . . . 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 6297	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) G ( A - B ) ) x. t ) = ( 0 x. t ) )
60	42	mul02d 9773	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( 0 x. t ) = 0 )
61	56, 59, 60	3eqtrd 2512	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( ( A - B ) x. t ) ) = 0 )
62	61	negeqd 9810	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> -u ( ( A - B ) G ( ( A - B ) x. t ) ) = -u 0 )
63		neg0 9861	. . . . . 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 2512	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) x. t ) G ( A - B ) ) = 0 )
66	37, 51, 65	3eqtrd 2512	. . 3 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 )
67	66	rexlimdv3a 2957	. 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 973   = wceq 1379   e. wcel 1767   E.wrex 2815   ` cfv 5586  (class class class)co 6282   |-> cmpt2 6284   CCcc 9486   RRcr 9487   0cc0 9488   + caddc 9491   x. cmul 9493   - cmin 9801   -ucneg 9802   / cdiv 10202   *ccj 12886   Imcim 12888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-2 10590  df-cj 12889  df-re 12890  df-im 12891

This theorem is referenced by: (None)