Theorem sigardiv 41050

Description: If signed area between vectors B - A and C - A is zero, then those vectors lie on the same line. (Contributed by Saveliy Skresanov, 22-Sep-2017.)

Hypotheses

Ref	Expression
sigar	|- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
sigardiv.a	|- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
sigardiv.b	|- ( ph -> -. C = A )
sigardiv.c	|- ( ph -> ( ( B - A ) G ( C - A ) ) = 0 )

Assertion

Ref	Expression
sigardiv	|- ( ph -> ( ( B - A ) / ( C - A ) ) e. RR )

Distinct variable groups:   x, y, A   x, B, y   x, C, y

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

Proof of Theorem sigardiv

Step	Hyp	Ref	Expression
1		sigardiv.a	. . . . . . . 8 |- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
2	1	simp2d 1074	. . . . . . 7 |- ( ph -> B e. CC )
3	1	simp1d 1073	. . . . . . 7 |- ( ph -> A e. CC )
4	2, 3	subcld 10392	. . . . . 6 |- ( ph -> ( B - A ) e. CC )
5	1	simp3d 1075	. . . . . . 7 |- ( ph -> C e. CC )
6	5, 3	subcld 10392	. . . . . 6 |- ( ph -> ( C - A ) e. CC )
7		sigardiv.b	. . . . . . . 8 |- ( ph -> -. C = A )
8	7	neqned 2801	. . . . . . 7 |- ( ph -> C =/= A )
9	5, 3, 8	subne0d 10401	. . . . . 6 |- ( ph -> ( C - A ) =/= 0 )
10	4, 6, 9	cjdivd 13963	. . . . 5 |- ( ph -> ( * ` ( ( B - A ) / ( C - A ) ) ) = ( ( * ` ( B - A ) ) / ( * ` ( C - A ) ) ) )
11	4	cjcld 13936	. . . . . . 7 |- ( ph -> ( * ` ( B - A ) ) e. CC )
12	6	cjcld 13936	. . . . . . 7 |- ( ph -> ( * ` ( C - A ) ) e. CC )
13	6, 9	cjne0d 13943	. . . . . . 7 |- ( ph -> ( * ` ( C - A ) ) =/= 0 )
14	11, 12, 6, 13, 9	divcan5rd 10828	. . . . . 6 |- ( ph -> ( ( ( * ` ( B - A ) ) x. ( C - A ) ) / ( ( * ` ( C - A ) ) x. ( C - A ) ) ) = ( ( * ` ( B - A ) ) / ( * ` ( C - A ) ) ) )
15	11, 6	mulcld 10060	. . . . . . . 8 |- ( ph -> ( ( * ` ( B - A ) ) x. ( C - A ) ) e. CC )
16		sigar	. . . . . . . . . . 11 |- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
17	16	sigarval 41039	. . . . . . . . . 10 |- ( ( ( B - A ) e. CC /\ ( C - A ) e. CC ) -> ( ( B - A ) G ( C - A ) ) = ( Im ` ( ( * ` ( B - A ) ) x. ( C - A ) ) ) )
18	4, 6, 17	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( ( B - A ) G ( C - A ) ) = ( Im ` ( ( * ` ( B - A ) ) x. ( C - A ) ) ) )
19		sigardiv.c	. . . . . . . . 9 |- ( ph -> ( ( B - A ) G ( C - A ) ) = 0 )
20	18, 19	eqtr3d 2658	. . . . . . . 8 |- ( ph -> ( Im ` ( ( * ` ( B - A ) ) x. ( C - A ) ) ) = 0 )
21	15, 20	reim0bd 13940	. . . . . . 7 |- ( ph -> ( ( * ` ( B - A ) ) x. ( C - A ) ) e. RR )
22	6, 12	mulcomd 10061	. . . . . . . 8 |- ( ph -> ( ( C - A ) x. ( * ` ( C - A ) ) ) = ( ( * ` ( C - A ) ) x. ( C - A ) ) )
23	6	cjmulrcld 13946	. . . . . . . 8 |- ( ph -> ( ( C - A ) x. ( * ` ( C - A ) ) ) e. RR )
24	22, 23	eqeltrrd 2702	. . . . . . 7 |- ( ph -> ( ( * ` ( C - A ) ) x. ( C - A ) ) e. RR )
25	12, 6, 13, 9	mulne0d 10679	. . . . . . 7 |- ( ph -> ( ( * ` ( C - A ) ) x. ( C - A ) ) =/= 0 )
26	21, 24, 25	redivcld 10853	. . . . . 6 |- ( ph -> ( ( ( * ` ( B - A ) ) x. ( C - A ) ) / ( ( * ` ( C - A ) ) x. ( C - A ) ) ) e. RR )
27	14, 26	eqeltrrd 2702	. . . . 5 |- ( ph -> ( ( * ` ( B - A ) ) / ( * ` ( C - A ) ) ) e. RR )
28	10, 27	eqeltrd 2701	. . . 4 |- ( ph -> ( * ` ( ( B - A ) / ( C - A ) ) ) e. RR )
29	28	cjred 13966	. . 3 |- ( ph -> ( * ` ( * ` ( ( B - A ) / ( C - A ) ) ) ) = ( * ` ( ( B - A ) / ( C - A ) ) ) )
30	4, 6, 9	divcld 10801	. . . 4 |- ( ph -> ( ( B - A ) / ( C - A ) ) e. CC )
31	30	cjcjd 13939	. . 3 |- ( ph -> ( * ` ( * ` ( ( B - A ) / ( C - A ) ) ) ) = ( ( B - A ) / ( C - A ) ) )
32	29, 31	eqtr3d 2658	. 2 |- ( ph -> ( * ` ( ( B - A ) / ( C - A ) ) ) = ( ( B - A ) / ( C - A ) ) )
33	32, 28	eqeltrrd 2702	1 |- ( ph -> ( ( B - A ) / ( C - A ) ) e. RR )

Syntax hints:   -. wn 3   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   x. cmul 9941   - cmin 10266   / 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:  sigarcol  41053  sharhght  41054