Theorem sigarcol 41053

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 1074	. . . . . . 7 |- ( ph -> B e. CC )
4	2	simp3d 1075	. . . . . . 7 |- ( ph -> C e. CC )
5	2	simp1d 1073	. . . . . . 7 |- ( ph -> A e. CC )
6	3, 4, 5	3jca 1242	. . . . . 6 |- ( ph -> ( B e. CC /\ C e. CC /\ A e. CC ) )
7	6	adantr 481	. . . . 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 481	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> -. A = B )
10	1	sigarperm 41049	. . . . . . . . 9 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - C ) G ( B - C ) ) = ( ( B - A ) G ( C - A ) ) )
11	2, 10	syl 17	. . . . . . . 8 |- ( ph -> ( ( A - C ) G ( B - C ) ) = ( ( B - A ) G ( C - A ) ) )
12	1	sigarperm 41049	. . . . . . . . 9 |- ( ( B e. CC /\ C e. CC /\ A e. CC ) -> ( ( B - A ) G ( C - A ) ) = ( ( C - B ) G ( A - B ) ) )
13	6, 12	syl 17	. . . . . . . 8 |- ( ph -> ( ( B - A ) G ( C - A ) ) = ( ( C - B ) G ( A - B ) ) )
14	11, 13	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
15	14	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 <-> ( ( C - B ) G ( A - B ) ) = 0 ) )
16	15	biimpa 501	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) G ( A - B ) ) = 0 )
17	1, 7, 9, 16	sigardiv 41050	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) / ( A - B ) ) e. RR )
18	4, 3	subcld 10392	. . . . . . . 8 |- ( ph -> ( C - B ) e. CC )
19	18	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( C - B ) e. CC )
20	5, 3	subcld 10392	. . . . . . . 8 |- ( ph -> ( A - B ) e. CC )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) e. CC )
22	5	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A e. CC )
23	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> B e. CC )
24	9	neqned 2801	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A =/= B )
25	22, 23, 24	subne0d 10401	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) =/= 0 )
26	19, 21, 25	divcan1d 10802	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) = ( C - B ) )
27	26	oveq2d 6666	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) = ( B + ( C - B ) ) )
28	4	adantr 481	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C e. CC )
29	23, 28	pncan3d 10395	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( C - B ) ) = C )
30	27, 29	eqtr2d 2657	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
31		oveq1 6657	. . . . . . 7 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( t x. ( A - B ) ) = ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) )
32	31	oveq2d 6666	. . . . . 6 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( B + ( t x. ( A - B ) ) ) = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
33	32	eqeq2d 2632	. . . . 5 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( C = ( B + ( t x. ( A - B ) ) ) <-> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) ) )
34	33	rspcev 3309	. . . 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 693	. . 3 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) )
36	35	ex 450	. 2 |- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 -> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) ) )
37	14	3ad2ant1 1082	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
38		simp3 1063	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> C = ( B + ( t x. ( A - B ) ) ) )
39	38	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( ( B + ( t x. ( A - B ) ) ) - B ) )
40	3	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> B e. CC )
41		simp2 1062	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. RR )
42	41	recnd 10068	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. CC )
43	5	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> A e. CC )
44	43, 40	subcld 10392	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( A - B ) e. CC )
45	42, 44	mulcld 10060	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) e. CC )
46	40, 45	pncan2d 10394	. . . . . . 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 2656	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( t x. ( A - B ) ) )
48	47	oveq1d 6665	. . . . 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 10061	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) = ( ( A - B ) x. t ) )
50	49	oveq1d 6665	. . . . 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 2656	. . . 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 10060	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) x. t ) e. CC )
53	1	sigarac 41041	. . . . . 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 693	. . . . 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 41046	. . . . . . . 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 1326	. . . . . . 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 41047	. . . . . . . . 9 |- ( ( A - B ) e. CC -> ( ( A - B ) G ( A - B ) ) = 0 )
58	44, 57	syl 17	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( A - B ) ) = 0 )
59	58	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) G ( A - B ) ) x. t ) = ( 0 x. t ) )
60	42	mul02d 10234	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( 0 x. t ) = 0 )
61	56, 59, 60	3eqtrd 2660	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( ( A - B ) x. t ) ) = 0 )
62	61	negeqd 10275	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> -u ( ( A - B ) G ( ( A - B ) x. t ) ) = -u 0 )
63		neg0 10327	. . . . . 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 2660	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) x. t ) G ( A - B ) ) = 0 )
66	37, 51, 65	3eqtrd 2660	. . 3 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 )
67	66	rexlimdv3a 3033	. 2 |- ( ph -> ( E. t e. RR C = ( B + ( t x. ( A - B ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 ) )
68	36, 67	impbid 202	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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   *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: (None)