Theorem bj-bary1lem1 33161

Description: Existence and uniqueness (and actual computation) of barycentric coordinates in dimension 1 (complex line). (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-bary1lem1	|- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> T = ( ( X - A ) / ( B - A ) ) ) )

Proof of Theorem bj-bary1lem1

Step	Hyp	Ref	Expression
1		bj-bary1.s	. . . . . . 7 |- ( ph -> S e. CC )
2		bj-bary1.t	. . . . . . 7 |- ( ph -> T e. CC )
3	1, 2	pncand 10393	. . . . . 6 |- ( ph -> ( ( S + T ) - T ) = S )
4		oveq1 6657	. . . . . 6 |- ( ( S + T ) = 1 -> ( ( S + T ) - T ) = ( 1 - T ) )
5		pm5.31 612	. . . . . 6 |- ( ( ( ( S + T ) - T ) = S /\ ( ( S + T ) = 1 -> ( ( S + T ) - T ) = ( 1 - T ) ) ) -> ( ( S + T ) = 1 -> ( ( ( S + T ) - T ) = ( 1 - T ) /\ ( ( S + T ) - T ) = S ) ) )
6	3, 4, 5	sylancl 694	. . . . 5 |- ( ph -> ( ( S + T ) = 1 -> ( ( ( S + T ) - T ) = ( 1 - T ) /\ ( ( S + T ) - T ) = S ) ) )
7		eqtr2 2642	. . . . . 6 |- ( ( ( ( S + T ) - T ) = ( 1 - T ) /\ ( ( S + T ) - T ) = S ) -> ( 1 - T ) = S )
8	7	eqcomd 2628	. . . . 5 |- ( ( ( ( S + T ) - T ) = ( 1 - T ) /\ ( ( S + T ) - T ) = S ) -> S = ( 1 - T ) )
9	6, 8	syl6 35	. . . 4 |- ( ph -> ( ( S + T ) = 1 -> S = ( 1 - T ) ) )
10		oveq1 6657	. . . . . . . 8 |- ( S = ( 1 - T ) -> ( S x. A ) = ( ( 1 - T ) x. A ) )
11	10	oveq1d 6665	. . . . . . 7 |- ( S = ( 1 - T ) -> ( ( S x. A ) + ( T x. B ) ) = ( ( ( 1 - T ) x. A ) + ( T x. B ) ) )
12		eqtr 2641	. . . . . . 7 |- ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( ( S x. A ) + ( T x. B ) ) = ( ( ( 1 - T ) x. A ) + ( T x. B ) ) ) -> X = ( ( ( 1 - T ) x. A ) + ( T x. B ) ) )
13	11, 12	sylan2 491	. . . . . 6 |- ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ S = ( 1 - T ) ) -> X = ( ( ( 1 - T ) x. A ) + ( T x. B ) ) )
14		1cnd 10056	. . . . . . . . 9 |- ( ph -> 1 e. CC )
15		bj-bary1.a	. . . . . . . . 9 |- ( ph -> A e. CC )
16	14, 2, 15	subdird 10487	. . . . . . . 8 |- ( ph -> ( ( 1 - T ) x. A ) = ( ( 1 x. A ) - ( T x. A ) ) )
17	15	mulid2d 10058	. . . . . . . . 9 |- ( ph -> ( 1 x. A ) = A )
18	17	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( 1 x. A ) - ( T x. A ) ) = ( A - ( T x. A ) ) )
19	16, 18	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1 - T ) x. A ) = ( A - ( T x. A ) ) )
20	19	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) = ( ( A - ( T x. A ) ) + ( T x. B ) ) )
21	13, 20	sylan9eqr 2678	. . . . 5 |- ( ( ph /\ ( X = ( ( S x. A ) + ( T x. B ) ) /\ S = ( 1 - T ) ) ) -> X = ( ( A - ( T x. A ) ) + ( T x. B ) ) )
22	21	ex 450	. . . 4 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ S = ( 1 - T ) ) -> X = ( ( A - ( T x. A ) ) + ( T x. B ) ) ) )
23	9, 22	sylan2d 499	. . 3 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> X = ( ( A - ( T x. A ) ) + ( T x. B ) ) ) )
24	2, 15	mulcld 10060	. . . . . 6 |- ( ph -> ( T x. A ) e. CC )
25		bj-bary1.b	. . . . . . 7 |- ( ph -> B e. CC )
26	2, 25	mulcld 10060	. . . . . 6 |- ( ph -> ( T x. B ) e. CC )
27	15, 24, 26	subadd23d 10414	. . . . 5 |- ( ph -> ( ( A - ( T x. A ) ) + ( T x. B ) ) = ( A + ( ( T x. B ) - ( T x. A ) ) ) )
28	2, 25, 15	subdid 10486	. . . . . . 7 |- ( ph -> ( T x. ( B - A ) ) = ( ( T x. B ) - ( T x. A ) ) )
29	28	eqcomd 2628	. . . . . 6 |- ( ph -> ( ( T x. B ) - ( T x. A ) ) = ( T x. ( B - A ) ) )
30	29	oveq2d 6666	. . . . 5 |- ( ph -> ( A + ( ( T x. B ) - ( T x. A ) ) ) = ( A + ( T x. ( B - A ) ) ) )
31	27, 30	eqtrd 2656	. . . 4 |- ( ph -> ( ( A - ( T x. A ) ) + ( T x. B ) ) = ( A + ( T x. ( B - A ) ) ) )
32	31	eqeq2d 2632	. . 3 |- ( ph -> ( X = ( ( A - ( T x. A ) ) + ( T x. B ) ) <-> X = ( A + ( T x. ( B - A ) ) ) ) )
33	23, 32	sylibd 229	. 2 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> X = ( A + ( T x. ( B - A ) ) ) ) )
34		oveq1 6657	. . 3 |- ( X = ( A + ( T x. ( B - A ) ) ) -> ( X - A ) = ( ( A + ( T x. ( B - A ) ) ) - A ) )
35	25, 15	subcld 10392	. . . . . 6 |- ( ph -> ( B - A ) e. CC )
36	2, 35	mulcld 10060	. . . . 5 |- ( ph -> ( T x. ( B - A ) ) e. CC )
37	15, 36	pncan2d 10394	. . . 4 |- ( ph -> ( ( A + ( T x. ( B - A ) ) ) - A ) = ( T x. ( B - A ) ) )
38	37	eqeq2d 2632	. . 3 |- ( ph -> ( ( X - A ) = ( ( A + ( T x. ( B - A ) ) ) - A ) <-> ( X - A ) = ( T x. ( B - A ) ) ) )
39	34, 38	syl5ib 234	. 2 |- ( ph -> ( X = ( A + ( T x. ( B - A ) ) ) -> ( X - A ) = ( T x. ( B - A ) ) ) )
40		eqcom 2629	. . 3 |- ( ( X - A ) = ( T x. ( B - A ) ) <-> ( T x. ( B - A ) ) = ( X - A ) )
41	2, 35	mulcomd 10061	. . . . 5 |- ( ph -> ( T x. ( B - A ) ) = ( ( B - A ) x. T ) )
42	41	eqeq1d 2624	. . . 4 |- ( ph -> ( ( T x. ( B - A ) ) = ( X - A ) <-> ( ( B - A ) x. T ) = ( X - A ) ) )
43		bj-bary1.x	. . . . . . 7 |- ( ph -> X e. CC )
44	43, 15	subcld 10392	. . . . . 6 |- ( ph -> ( X - A ) e. CC )
45		bj-bary1.neq	. . . . . . . 8 |- ( ph -> A =/= B )
46	45	necomd 2849	. . . . . . 7 |- ( ph -> B =/= A )
47	25, 15, 46	subne0d 10401	. . . . . 6 |- ( ph -> ( B - A ) =/= 0 )
48	35, 2, 44, 47	bj-rdiv 33156	. . . . 5 |- ( ph -> ( ( ( B - A ) x. T ) = ( X - A ) <-> T = ( ( X - A ) / ( B - A ) ) ) )
49	48	biimpd 219	. . . 4 |- ( ph -> ( ( ( B - A ) x. T ) = ( X - A ) -> T = ( ( X - A ) / ( B - A ) ) ) )
50	42, 49	sylbid 230	. . 3 |- ( ph -> ( ( T x. ( B - A ) ) = ( X - A ) -> T = ( ( X - A ) / ( B - A ) ) ) )
51	40, 50	syl5bi 232	. 2 |- ( ph -> ( ( X - A ) = ( T x. ( B - A ) ) -> T = ( ( X - A ) / ( B - A ) ) ) )
52	33, 39, 51	3syld 60	1 |- ( ph -> ( ( X = ( ( S x. A ) + ( T x. B ) ) /\ ( S + T ) = 1 ) -> T = ( ( X - A ) / ( B - A ) ) ) )

Syntax hints:   -> wi 4   /\ 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:  bj-bary1  33162