Theorem bj-bary1 33162

Description: Barycentric coordinates in one dimension. (Contributed by BJ, 6-Jun-2019.)

Hypotheses

Ref	Expression
bj-bary1.a	|- ( ph -> A e. CC )
bj-bary1.b	|- ( ph -> B e. CC )
bj-bary1.x	|- ( ph -> X e. CC )
bj-bary1.neq	|- ( ph -> A =/= B )
bj-bary1.s	|- ( ph -> S e. CC )
bj-bary1.t	|- ( ph -> T e. CC )

Assertion

Ref	Expression
bj-bary1	|- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) <-> ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) ) )

Proof of Theorem bj-bary1

Step	Hyp	Ref	Expression
1		bj-bary1.s	. . . . . . . . 9 |- ( ph -> S e. CC )
2		bj-bary1.a	. . . . . . . . 9 |- ( ph -> A e. CC )
3	1, 2	mulcld 10060	. . . . . . . 8 |- ( ph -> ( S x. A ) e. CC )
4		bj-bary1.t	. . . . . . . . 9 |- ( ph -> T e. CC )
5		bj-bary1.b	. . . . . . . . 9 |- ( ph -> B e. CC )
6	4, 5	mulcld 10060	. . . . . . . 8 |- ( ph -> ( T x. B ) e. CC )
7	3, 6	addcomd 10238	. . . . . . 7 |- ( ph -> ( ( S x. A ) + ( T x. B ) ) = ( ( T x. B ) + ( S x. A ) ) )
8	7	eqeq2d 2632	. . . . . 6 |- ( ph -> ( X = ( ( S x. A ) + ( T x. B ) ) <-> X = ( ( T x. B ) + ( S x. A ) ) ) )
9	8	biimpd 219	. . . . 5 |- ( ph -> ( X = ( ( S x. A ) + ( T x. B ) ) -> X = ( ( T x. B ) + ( S x. A ) ) ) )
10	1, 4	addcomd 10238	. . . . . . 7 |- ( ph -> ( S + T ) = ( T + S ) )
11	10	eqeq1d 2624	. . . . . 6 |- ( ph -> ( ( S + T ) = 1 <-> ( T + S ) = 1 ) )
12	11	biimpd 219	. . . . 5 |- ( ph -> ( ( S + T ) = 1 -> ( T + S ) = 1 ) )
13		bj-bary1.x	. . . . . 6 |- ( ph -> X e. CC )
14		bj-bary1.neq	. . . . . . 7 |- ( ph -> A =/= B )
15	14	necomd 2849	. . . . . 6 |- ( ph -> B =/= A )
16	5, 2, 13, 15, 4, 1	bj-bary1lem1 33161	. . . . 5 |- ( ph -> ( ( X = ( ( T x. B ) + ( S x. A ) ) /\ ( T + S ) = 1 ) -> S = ( ( X - B ) / ( A - B ) ) ) )
17	9, 12, 16	syl2and 500	. . . 4 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> S = ( ( X - B ) / ( A - B ) ) ) )
18	13, 5, 2, 5, 14	div2subd 10851	. . . . 5 |- ( ph -> ( ( X - B ) / ( A - B ) ) = ( ( B - X ) / ( B - A ) ) )
19	18	eqeq2d 2632	. . . 4 |- ( ph -> ( S = ( ( X - B ) / ( A - B ) ) <-> S = ( ( B - X ) / ( B - A ) ) ) )
20	17, 19	sylibd 229	. . 3 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> S = ( ( B - X ) / ( B - A ) ) ) )
21	2, 5, 13, 14, 1, 4	bj-bary1lem1 33161	. . 3 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> T = ( ( X - A ) / ( B - A ) ) ) )
22	20, 21	jcad 555	. 2 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) ) )
23	2, 5, 13, 14	bj-bary1lem 33160	. . . 4 |- ( ph -> X = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) )
24		oveq1 6657	. . . . . 6 |- ( S = ( ( B - X ) / ( B - A ) ) -> ( S x. A ) = ( ( ( B - X ) / ( B - A ) ) x. A ) )
25		oveq1 6657	. . . . . 6 |- ( T = ( ( X - A ) / ( B - A ) ) -> ( T x. B ) = ( ( ( X - A ) / ( B - A ) ) x. B ) )
26	24, 25	oveqan12d 6669	. . . . 5 |- ( ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) -> ( ( S x. A ) + ( T x. B ) ) = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) )
27	26	a1i 11	. . . 4 |- ( ph -> ( ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) -> ( ( S x. A ) + ( T x. B ) ) = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) ) )
28		eqtr3 2643	. . . 4 |- ( ( X = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) /\ ( ( S x. A ) + ( T x. B ) ) = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) ) -> X = ( ( S x. A ) + ( T x. B ) ) )
29	23, 27, 28	syl6an 568	. . 3 |- ( ph -> ( ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) -> X = ( ( S x. A ) + ( T x. B ) ) ) )
30		oveq12 6659	. . . 4 |- ( ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) -> ( S + T ) = ( ( ( B - X ) / ( B - A ) ) + ( ( X - A ) / ( B - A ) ) ) )
31	5, 13	subcld 10392	. . . . . . 7 |- ( ph -> ( B - X ) e. CC )
32	13, 2	subcld 10392	. . . . . . 7 |- ( ph -> ( X - A ) e. CC )
33	5, 2	subcld 10392	. . . . . . 7 |- ( ph -> ( B - A ) e. CC )
34	5, 2, 15	subne0d 10401	. . . . . . 7 |- ( ph -> ( B - A ) =/= 0 )
35	31, 32, 33, 34	divdird 10839	. . . . . 6 |- ( ph -> ( ( ( B - X ) + ( X - A ) ) / ( B - A ) ) = ( ( ( B - X ) / ( B - A ) ) + ( ( X - A ) / ( B - A ) ) ) )
36	5, 13, 2	npncand 10416	. . . . . . 7 |- ( ph -> ( ( B - X ) + ( X - A ) ) = ( B - A ) )
37	33, 34, 36	diveq1bd 10849	. . . . . 6 |- ( ph -> ( ( ( B - X ) + ( X - A ) ) / ( B - A ) ) = 1 )
38	35, 37	eqtr3d 2658	. . . . 5 |- ( ph -> ( ( ( B - X ) / ( B - A ) ) + ( ( X - A ) / ( B - A ) ) ) = 1 )
39	38	eqeq2d 2632	. . . 4 |- ( ph -> ( ( S + T ) = ( ( ( B - X ) / ( B - A ) ) + ( ( X - A ) / ( B - A ) ) ) <-> ( S + T ) = 1 ) )
40	30, 39	syl5ib 234	. . 3 |- ( ph -> ( ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) -> ( S + T ) = 1 ) )
41	29, 40	jcad 555	. 2 |- ( ph -> ( ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) -> ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) ) )
42	22, 41	impbid 202	1 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) <-> ( S = ( ( B - X ) / ( B - A ) ) /\ T = ( ( X - A ) / ( B - A ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684

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

This theorem is referenced by: (None)