Theorem sigarcol 38619

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 1043	. . . . . . 7 |- ( ph -> B e. CC )
4	2	simp3d 1044	. . . . . . 7 |- ( ph -> C e. CC )
5	2	simp1d 1042	. . . . . . 7 |- ( ph -> A e. CC )
6	3, 4, 5	3jca 1210	. . . . . 6 |- ( ph -> ( B e. CC /\ C e. CC /\ A e. CC ) )
7	6	adantr 472	. . . . 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 472	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> -. A = B )
10	1	sigarperm 38615	. . . . . . . . 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 38615	. . . . . . . . 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 2505	. . . . . . 7 |- ( ph -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
15	14	eqeq1d 2473	. . . . . 6 |- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 <-> ( ( C - B ) G ( A - B ) ) = 0 ) )
16	15	biimpa 492	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) G ( A - B ) ) = 0 )
17	1, 7, 9, 16	sigardiv 38616	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( C - B ) / ( A - B ) ) e. RR )
18	4, 3	subcld 10005	. . . . . . . 8 |- ( ph -> ( C - B ) e. CC )
19	18	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( C - B ) e. CC )
20	5, 3	subcld 10005	. . . . . . . 8 |- ( ph -> ( A - B ) e. CC )
21	20	adantr 472	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) e. CC )
22	5	adantr 472	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A e. CC )
23	3	adantr 472	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> B e. CC )
24	9	neqned 2650	. . . . . . . 8 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> A =/= B )
25	22, 23, 24	subne0d 10014	. . . . . . 7 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( A - B ) =/= 0 )
26	19, 21, 25	divcan1d 10406	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) = ( C - B ) )
27	26	oveq2d 6324	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) = ( B + ( C - B ) ) )
28	4	adantr 472	. . . . . 6 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C e. CC )
29	23, 28	pncan3d 10008	. . . . 5 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> ( B + ( C - B ) ) = C )
30	27, 29	eqtr2d 2506	. . . 4 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> C = ( B + ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) ) )
31		oveq1 6315	. . . . . . 7 |- ( t = ( ( C - B ) / ( A - B ) ) -> ( t x. ( A - B ) ) = ( ( ( C - B ) / ( A - B ) ) x. ( A - B ) ) )
32	31	oveq2d 6324	. . . . . 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 3136	. . . 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 673	. . 3 |- ( ( ph /\ ( ( A - C ) G ( B - C ) ) = 0 ) -> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) )
36	35	ex 441	. 2 |- ( ph -> ( ( ( A - C ) G ( B - C ) ) = 0 -> E. t e. RR C = ( B + ( t x. ( A - B ) ) ) ) )
37	14	3ad2ant1 1051	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = ( ( C - B ) G ( A - B ) ) )
38		simp3 1032	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> C = ( B + ( t x. ( A - B ) ) ) )
39	38	oveq1d 6323	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( ( B + ( t x. ( A - B ) ) ) - B ) )
40	3	3ad2ant1 1051	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> B e. CC )
41		simp2 1031	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. RR )
42	41	recnd 9687	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> t e. CC )
43	5	3ad2ant1 1051	. . . . . . . . . 10 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> A e. CC )
44	43, 40	subcld 10005	. . . . . . . . 9 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( A - B ) e. CC )
45	42, 44	mulcld 9681	. . . . . . . 8 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) e. CC )
46	40, 45	pncan2d 10007	. . . . . . 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 2505	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( C - B ) = ( t x. ( A - B ) ) )
48	47	oveq1d 6323	. . . . 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 9682	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( t x. ( A - B ) ) = ( ( A - B ) x. t ) )
50	49	oveq1d 6323	. . . . 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 2505	. . . 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 9681	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) x. t ) e. CC )
53	1	sigarac 38607	. . . . . 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 673	. . . . 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 38612	. . . . . . . 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 1292	. . . . . . 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 38613	. . . . . . . . 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 6323	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) G ( A - B ) ) x. t ) = ( 0 x. t ) )
60	42	mul02d 9849	. . . . . . 7 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( 0 x. t ) = 0 )
61	56, 59, 60	3eqtrd 2509	. . . . . 6 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - B ) G ( ( A - B ) x. t ) ) = 0 )
62	61	negeqd 9889	. . . . 5 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> -u ( ( A - B ) G ( ( A - B ) x. t ) ) = -u 0 )
63		neg0 9940	. . . . . 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 2509	. . . 4 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( ( A - B ) x. t ) G ( A - B ) ) = 0 )
66	37, 51, 65	3eqtrd 2509	. . 3 |- ( ( ph /\ t e. RR /\ C = ( B + ( t x. ( A - B ) ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 )
67	66	rexlimdv3a 2873	. 2 |- ( ph -> ( E. t e. RR C = ( B + ( t x. ( A - B ) ) ) -> ( ( A - C ) G ( B - C ) ) = 0 ) )
68	36, 67	impbid 195	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 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   E.wrex 2757   ` cfv 5589  (class class class)co 6308   |-> cmpt2 6310   CCcc 9555   RRcr 9556   0cc0 9557   + caddc 9560   x. cmul 9562   - cmin 9880   -ucneg 9881   / cdiv 10291   *ccj 13236   Imcim 13238

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-2 10690  df-cj 13239  df-re 13240  df-im 13241

This theorem is referenced by: (None)