Theorem sharhght 41054

Description: Let A B C be a triangle, and let D lie on the line A B. Then (doubled) areas of triangles A D C and C D B relate as lengths of corresponding bases A D and D B. (Contributed by Saveliy Skresanov, 23-Sep-2017.)

Hypotheses

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

Assertion

Ref	Expression
sharhght	|- ( ph -> ( ( ( C - A ) G ( D - A ) ) x. ( B - D ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( A - D ) ) )

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

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

Proof of Theorem sharhght

Step	Hyp	Ref	Expression
1		sharhght.a	. . . . . . . . 9 |- ( ph -> ( A e. CC /\ B e. CC /\ C e. CC ) )
2	1	simp3d 1075	. . . . . . . 8 |- ( ph -> C e. CC )
3	1	simp1d 1073	. . . . . . . 8 |- ( ph -> A e. CC )
4	2, 3	subcld 10392	. . . . . . 7 |- ( ph -> ( C - A ) e. CC )
5	4	adantr 481	. . . . . 6 |- ( ( ph /\ B = D ) -> ( C - A ) e. CC )
6		sharhght.b	. . . . . . . . 9 |- ( ph -> ( D e. CC /\ ( ( A - D ) G ( B - D ) ) = 0 ) )
7	6	simpld 475	. . . . . . . 8 |- ( ph -> D e. CC )
8	7, 3	subcld 10392	. . . . . . 7 |- ( ph -> ( D - A ) e. CC )
9	8	adantr 481	. . . . . 6 |- ( ( ph /\ B = D ) -> ( D - A ) e. CC )
10		sharhght.sigar	. . . . . . 7 |- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) )
11	10	sigarim 41040	. . . . . 6 |- ( ( ( C - A ) e. CC /\ ( D - A ) e. CC ) -> ( ( C - A ) G ( D - A ) ) e. RR )
12	5, 9, 11	syl2anc 693	. . . . 5 |- ( ( ph /\ B = D ) -> ( ( C - A ) G ( D - A ) ) e. RR )
13	12	recnd 10068	. . . 4 |- ( ( ph /\ B = D ) -> ( ( C - A ) G ( D - A ) ) e. CC )
14	13	mul01d 10235	. . 3 |- ( ( ph /\ B = D ) -> ( ( ( C - A ) G ( D - A ) ) x. 0 ) = 0 )
15	1	simp2d 1074	. . . . . 6 |- ( ph -> B e. CC )
16	15	adantr 481	. . . . 5 |- ( ( ph /\ B = D ) -> B e. CC )
17		simpr 477	. . . . 5 |- ( ( ph /\ B = D ) -> B = D )
18	16, 17	subeq0bd 10456	. . . 4 |- ( ( ph /\ B = D ) -> ( B - D ) = 0 )
19	18	oveq2d 6666	. . 3 |- ( ( ph /\ B = D ) -> ( ( ( C - A ) G ( D - A ) ) x. ( B - D ) ) = ( ( ( C - A ) G ( D - A ) ) x. 0 ) )
20	2, 15	subcld 10392	. . . . . . . 8 |- ( ph -> ( C - B ) e. CC )
21	20	adantr 481	. . . . . . 7 |- ( ( ph /\ B = D ) -> ( C - B ) e. CC )
22	7, 15	subcld 10392	. . . . . . . 8 |- ( ph -> ( D - B ) e. CC )
23	22	adantr 481	. . . . . . 7 |- ( ( ph /\ B = D ) -> ( D - B ) e. CC )
24	10	sigarval 41039	. . . . . . 7 |- ( ( ( C - B ) e. CC /\ ( D - B ) e. CC ) -> ( ( C - B ) G ( D - B ) ) = ( Im ` ( ( * ` ( C - B ) ) x. ( D - B ) ) ) )
25	21, 23, 24	syl2anc 693	. . . . . 6 |- ( ( ph /\ B = D ) -> ( ( C - B ) G ( D - B ) ) = ( Im ` ( ( * ` ( C - B ) ) x. ( D - B ) ) ) )
26	7	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ B = D ) -> D e. CC )
27	17	eqcomd 2628	. . . . . . . . . 10 |- ( ( ph /\ B = D ) -> D = B )
28	26, 27	subeq0bd 10456	. . . . . . . . 9 |- ( ( ph /\ B = D ) -> ( D - B ) = 0 )
29	28	oveq2d 6666	. . . . . . . 8 |- ( ( ph /\ B = D ) -> ( ( * ` ( C - B ) ) x. ( D - B ) ) = ( ( * ` ( C - B ) ) x. 0 ) )
30	21	cjcld 13936	. . . . . . . . 9 |- ( ( ph /\ B = D ) -> ( * ` ( C - B ) ) e. CC )
31	30	mul01d 10235	. . . . . . . 8 |- ( ( ph /\ B = D ) -> ( ( * ` ( C - B ) ) x. 0 ) = 0 )
32	29, 31	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ B = D ) -> ( ( * ` ( C - B ) ) x. ( D - B ) ) = 0 )
33	32	fveq2d 6195	. . . . . 6 |- ( ( ph /\ B = D ) -> ( Im ` ( ( * ` ( C - B ) ) x. ( D - B ) ) ) = ( Im ` 0 ) )
34		0red 10041	. . . . . . 7 |- ( ( ph /\ B = D ) -> 0 e. RR )
35	34	reim0d 13965	. . . . . 6 |- ( ( ph /\ B = D ) -> ( Im ` 0 ) = 0 )
36	25, 33, 35	3eqtrd 2660	. . . . 5 |- ( ( ph /\ B = D ) -> ( ( C - B ) G ( D - B ) ) = 0 )
37	36	oveq1d 6665	. . . 4 |- ( ( ph /\ B = D ) -> ( ( ( C - B ) G ( D - B ) ) x. ( A - D ) ) = ( 0 x. ( A - D ) ) )
38	3	adantr 481	. . . . . 6 |- ( ( ph /\ B = D ) -> A e. CC )
39	38, 26	subcld 10392	. . . . 5 |- ( ( ph /\ B = D ) -> ( A - D ) e. CC )
40	39	mul02d 10234	. . . 4 |- ( ( ph /\ B = D ) -> ( 0 x. ( A - D ) ) = 0 )
41	37, 40	eqtrd 2656	. . 3 |- ( ( ph /\ B = D ) -> ( ( ( C - B ) G ( D - B ) ) x. ( A - D ) ) = 0 )
42	14, 19, 41	3eqtr4d 2666	. 2 |- ( ( ph /\ B = D ) -> ( ( ( C - A ) G ( D - A ) ) x. ( B - D ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( A - D ) ) )
43	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. B = D ) -> C e. CC )
44	15	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. B = D ) -> B e. CC )
45	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. B = D ) -> A e. CC )
46	43, 44, 45	npncand 10416	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( ( C - B ) + ( B - A ) ) = ( C - A ) )
47	46	oveq1d 6665	. . . . . . 7 |- ( ( ph /\ -. B = D ) -> ( ( ( C - B ) + ( B - A ) ) G ( D - A ) ) = ( ( C - A ) G ( D - A ) ) )
48	43, 44	subcld 10392	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( C - B ) e. CC )
49	8	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( D - A ) e. CC )
50	44, 45	subcld 10392	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( B - A ) e. CC )
51	10	sigaraf 41042	. . . . . . . 8 |- ( ( ( C - B ) e. CC /\ ( D - A ) e. CC /\ ( B - A ) e. CC ) -> ( ( ( C - B ) + ( B - A ) ) G ( D - A ) ) = ( ( ( C - B ) G ( D - A ) ) + ( ( B - A ) G ( D - A ) ) ) )
52	48, 49, 50, 51	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ -. B = D ) -> ( ( ( C - B ) + ( B - A ) ) G ( D - A ) ) = ( ( ( C - B ) G ( D - A ) ) + ( ( B - A ) G ( D - A ) ) ) )
53	47, 52	eqtr3d 2658	. . . . . 6 |- ( ( ph /\ -. B = D ) -> ( ( C - A ) G ( D - A ) ) = ( ( ( C - B ) G ( D - A ) ) + ( ( B - A ) G ( D - A ) ) ) )
54	6	simprd 479	. . . . . . . . 9 |- ( ph -> ( ( A - D ) G ( B - D ) ) = 0 )
55	54	adantr 481	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( ( A - D ) G ( B - D ) ) = 0 )
56	7	adantr 481	. . . . . . . . 9 |- ( ( ph /\ -. B = D ) -> D e. CC )
57	10	sigarperm 41049	. . . . . . . . 9 |- ( ( A e. CC /\ B e. CC /\ D e. CC ) -> ( ( A - D ) G ( B - D ) ) = ( ( B - A ) G ( D - A ) ) )
58	45, 44, 56, 57	syl3anc 1326	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( ( A - D ) G ( B - D ) ) = ( ( B - A ) G ( D - A ) ) )
59	55, 58	eqtr3d 2658	. . . . . . 7 |- ( ( ph /\ -. B = D ) -> 0 = ( ( B - A ) G ( D - A ) ) )
60	59	oveq2d 6666	. . . . . 6 |- ( ( ph /\ -. B = D ) -> ( ( ( C - B ) G ( D - A ) ) + 0 ) = ( ( ( C - B ) G ( D - A ) ) + ( ( B - A ) G ( D - A ) ) ) )
61	10	sigarim 41040	. . . . . . . . 9 |- ( ( ( C - B ) e. CC /\ ( D - A ) e. CC ) -> ( ( C - B ) G ( D - A ) ) e. RR )
62	48, 49, 61	syl2anc 693	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( ( C - B ) G ( D - A ) ) e. RR )
63	62	recnd 10068	. . . . . . 7 |- ( ( ph /\ -. B = D ) -> ( ( C - B ) G ( D - A ) ) e. CC )
64	63	addid1d 10236	. . . . . 6 |- ( ( ph /\ -. B = D ) -> ( ( ( C - B ) G ( D - A ) ) + 0 ) = ( ( C - B ) G ( D - A ) ) )
65	53, 60, 64	3eqtr2d 2662	. . . . 5 |- ( ( ph /\ -. B = D ) -> ( ( C - A ) G ( D - A ) ) = ( ( C - B ) G ( D - A ) ) )
66	44, 56	negsubdi2d 10408	. . . . . . . . . . . 12 |- ( ( ph /\ -. B = D ) -> -u ( B - D ) = ( D - B ) )
67	66	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ph /\ -. B = D ) -> ( D - B ) = -u ( B - D ) )
68	67	oveq1d 6665	. . . . . . . . . 10 |- ( ( ph /\ -. B = D ) -> ( ( D - B ) / ( B - D ) ) = ( -u ( B - D ) / ( B - D ) ) )
69	44, 56	subcld 10392	. . . . . . . . . . 11 |- ( ( ph /\ -. B = D ) -> ( B - D ) e. CC )
70		simpr 477	. . . . . . . . . . . . 13 |- ( ( ph /\ -. B = D ) -> -. B = D )
71	70	neqned 2801	. . . . . . . . . . . 12 |- ( ( ph /\ -. B = D ) -> B =/= D )
72	44, 56, 71	subne0d 10401	. . . . . . . . . . 11 |- ( ( ph /\ -. B = D ) -> ( B - D ) =/= 0 )
73	69, 69, 72	divnegd 10814	. . . . . . . . . 10 |- ( ( ph /\ -. B = D ) -> -u ( ( B - D ) / ( B - D ) ) = ( -u ( B - D ) / ( B - D ) ) )
74	69, 72	dividd 10799	. . . . . . . . . . 11 |- ( ( ph /\ -. B = D ) -> ( ( B - D ) / ( B - D ) ) = 1 )
75	74	negeqd 10275	. . . . . . . . . 10 |- ( ( ph /\ -. B = D ) -> -u ( ( B - D ) / ( B - D ) ) = -u 1 )
76	68, 73, 75	3eqtr2d 2662	. . . . . . . . 9 |- ( ( ph /\ -. B = D ) -> ( ( D - B ) / ( B - D ) ) = -u 1 )
77	76	oveq1d 6665	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( ( ( D - B ) / ( B - D ) ) x. ( A - D ) ) = ( -u 1 x. ( A - D ) ) )
78	45, 56	subcld 10392	. . . . . . . . 9 |- ( ( ph /\ -. B = D ) -> ( A - D ) e. CC )
79	78	mulm1d 10482	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( -u 1 x. ( A - D ) ) = -u ( A - D ) )
80	45, 56	negsubdi2d 10408	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> -u ( A - D ) = ( D - A ) )
81	77, 79, 80	3eqtrd 2660	. . . . . . 7 |- ( ( ph /\ -. B = D ) -> ( ( ( D - B ) / ( B - D ) ) x. ( A - D ) ) = ( D - A ) )
82	56, 44	subcld 10392	. . . . . . . 8 |- ( ( ph /\ -. B = D ) -> ( D - B ) e. CC )
83	82, 69, 78, 72	div32d 10824	. . . . . . 7 |- ( ( ph /\ -. B = D ) -> ( ( ( D - B ) / ( B - D ) ) x. ( A - D ) ) = ( ( D - B ) x. ( ( A - D ) / ( B - D ) ) ) )
84	81, 83	eqtr3d 2658	. . . . . 6 |- ( ( ph /\ -. B = D ) -> ( D - A ) = ( ( D - B ) x. ( ( A - D ) / ( B - D ) ) ) )
85	84	oveq2d 6666	. . . . 5 |- ( ( ph /\ -. B = D ) -> ( ( C - B ) G ( D - A ) ) = ( ( C - B ) G ( ( D - B ) x. ( ( A - D ) / ( B - D ) ) ) ) )
86	56, 45, 44	3jca 1242	. . . . . . 7 |- ( ( ph /\ -. B = D ) -> ( D e. CC /\ A e. CC /\ B e. CC ) )
87	10, 86, 70, 55	sigardiv 41050	. . . . . 6 |- ( ( ph /\ -. B = D ) -> ( ( A - D ) / ( B - D ) ) e. RR )
88	10	sigarls 41046	. . . . . 6 |- ( ( ( C - B ) e. CC /\ ( D - B ) e. CC /\ ( ( A - D ) / ( B - D ) ) e. RR ) -> ( ( C - B ) G ( ( D - B ) x. ( ( A - D ) / ( B - D ) ) ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( ( A - D ) / ( B - D ) ) ) )
89	48, 82, 87, 88	syl3anc 1326	. . . . 5 |- ( ( ph /\ -. B = D ) -> ( ( C - B ) G ( ( D - B ) x. ( ( A - D ) / ( B - D ) ) ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( ( A - D ) / ( B - D ) ) ) )
90	65, 85, 89	3eqtrd 2660	. . . 4 |- ( ( ph /\ -. B = D ) -> ( ( C - A ) G ( D - A ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( ( A - D ) / ( B - D ) ) ) )
91	90	oveq1d 6665	. . 3 |- ( ( ph /\ -. B = D ) -> ( ( ( C - A ) G ( D - A ) ) x. ( B - D ) ) = ( ( ( ( C - B ) G ( D - B ) ) x. ( ( A - D ) / ( B - D ) ) ) x. ( B - D ) ) )
92	10	sigarim 41040	. . . . . 6 |- ( ( ( C - B ) e. CC /\ ( D - B ) e. CC ) -> ( ( C - B ) G ( D - B ) ) e. RR )
93	92	recnd 10068	. . . . 5 |- ( ( ( C - B ) e. CC /\ ( D - B ) e. CC ) -> ( ( C - B ) G ( D - B ) ) e. CC )
94	48, 82, 93	syl2anc 693	. . . 4 |- ( ( ph /\ -. B = D ) -> ( ( C - B ) G ( D - B ) ) e. CC )
95	78, 69, 72	divcld 10801	. . . 4 |- ( ( ph /\ -. B = D ) -> ( ( A - D ) / ( B - D ) ) e. CC )
96	94, 95, 69	mulassd 10063	. . 3 |- ( ( ph /\ -. B = D ) -> ( ( ( ( C - B ) G ( D - B ) ) x. ( ( A - D ) / ( B - D ) ) ) x. ( B - D ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( ( ( A - D ) / ( B - D ) ) x. ( B - D ) ) ) )
97	78, 69, 72	divcan1d 10802	. . . 4 |- ( ( ph /\ -. B = D ) -> ( ( ( A - D ) / ( B - D ) ) x. ( B - D ) ) = ( A - D ) )
98	97	oveq2d 6666	. . 3 |- ( ( ph /\ -. B = D ) -> ( ( ( C - B ) G ( D - B ) ) x. ( ( ( A - D ) / ( B - D ) ) x. ( B - D ) ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( A - D ) ) )
99	91, 96, 98	3eqtrd 2660	. 2 |- ( ( ph /\ -. B = D ) -> ( ( ( C - A ) G ( D - A ) ) x. ( B - D ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( A - D ) ) )
100	42, 99	pm2.61dan 832	1 |- ( ph -> ( ( ( C - A ) G ( D - A ) ) x. ( B - D ) ) = ( ( ( C - B ) G ( D - B ) ) x. ( A - D ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + 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:  cevathlem2  41057