Theorem bj-bary1lem 33160

Description: A lemma for 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 )

Assertion

Ref	Expression
bj-bary1lem	|- ( ph -> X = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) )

Proof of Theorem bj-bary1lem

Step	Hyp	Ref	Expression
1		bj-bary1.b	. . . . . . . . . 10 |- ( ph -> B e. CC )
2		bj-bary1.a	. . . . . . . . . 10 |- ( ph -> A e. CC )
3	1, 2	mulcld 10060	. . . . . . . . 9 |- ( ph -> ( B x. A ) e. CC )
4		bj-bary1.x	. . . . . . . . . 10 |- ( ph -> X e. CC )
5	4, 2	mulcld 10060	. . . . . . . . 9 |- ( ph -> ( X x. A ) e. CC )
6	3, 5	subcld 10392	. . . . . . . 8 |- ( ph -> ( ( B x. A ) - ( X x. A ) ) e. CC )
7	4, 1	mulcld 10060	. . . . . . . 8 |- ( ph -> ( X x. B ) e. CC )
8	2, 1	mulcld 10060	. . . . . . . 8 |- ( ph -> ( A x. B ) e. CC )
9	6, 7, 8	addsub12d 10415	. . . . . . 7 |- ( ph -> ( ( ( B x. A ) - ( X x. A ) ) + ( ( X x. B ) - ( A x. B ) ) ) = ( ( X x. B ) + ( ( ( B x. A ) - ( X x. A ) ) - ( A x. B ) ) ) )
10	3, 5, 8	sub32d 10424	. . . . . . . . 9 |- ( ph -> ( ( ( B x. A ) - ( X x. A ) ) - ( A x. B ) ) = ( ( ( B x. A ) - ( A x. B ) ) - ( X x. A ) ) )
11	1, 2	bj-subcom 33154	. . . . . . . . . 10 |- ( ph -> ( ( B x. A ) - ( A x. B ) ) = 0 )
12	11	oveq1d 6665	. . . . . . . . 9 |- ( ph -> ( ( ( B x. A ) - ( A x. B ) ) - ( X x. A ) ) = ( 0 - ( X x. A ) ) )
13	10, 12	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( ( B x. A ) - ( X x. A ) ) - ( A x. B ) ) = ( 0 - ( X x. A ) ) )
14	13	oveq2d 6666	. . . . . . 7 |- ( ph -> ( ( X x. B ) + ( ( ( B x. A ) - ( X x. A ) ) - ( A x. B ) ) ) = ( ( X x. B ) + ( 0 - ( X x. A ) ) ) )
15	9, 14	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( ( B x. A ) - ( X x. A ) ) + ( ( X x. B ) - ( A x. B ) ) ) = ( ( X x. B ) + ( 0 - ( X x. A ) ) ) )
16		0cnd 10033	. . . . . . 7 |- ( ph -> 0 e. CC )
17	7, 16, 5	addsubassd 10412	. . . . . 6 |- ( ph -> ( ( ( X x. B ) + 0 ) - ( X x. A ) ) = ( ( X x. B ) + ( 0 - ( X x. A ) ) ) )
18	7	addid1d 10236	. . . . . . 7 |- ( ph -> ( ( X x. B ) + 0 ) = ( X x. B ) )
19	18	oveq1d 6665	. . . . . 6 |- ( ph -> ( ( ( X x. B ) + 0 ) - ( X x. A ) ) = ( ( X x. B ) - ( X x. A ) ) )
20	15, 17, 19	3eqtr2d 2662	. . . . 5 |- ( ph -> ( ( ( B x. A ) - ( X x. A ) ) + ( ( X x. B ) - ( A x. B ) ) ) = ( ( X x. B ) - ( X x. A ) ) )
21	1, 4, 2	subdird 10487	. . . . . 6 |- ( ph -> ( ( B - X ) x. A ) = ( ( B x. A ) - ( X x. A ) ) )
22	4, 2, 1	subdird 10487	. . . . . 6 |- ( ph -> ( ( X - A ) x. B ) = ( ( X x. B ) - ( A x. B ) ) )
23	21, 22	oveq12d 6668	. . . . 5 |- ( ph -> ( ( ( B - X ) x. A ) + ( ( X - A ) x. B ) ) = ( ( ( B x. A ) - ( X x. A ) ) + ( ( X x. B ) - ( A x. B ) ) ) )
24	4, 1, 2	subdid 10486	. . . . 5 |- ( ph -> ( X x. ( B - A ) ) = ( ( X x. B ) - ( X x. A ) ) )
25	20, 23, 24	3eqtr4rd 2667	. . . 4 |- ( ph -> ( X x. ( B - A ) ) = ( ( ( B - X ) x. A ) + ( ( X - A ) x. B ) ) )
26	25	oveq1d 6665	. . 3 |- ( ph -> ( ( X x. ( B - A ) ) / ( B - A ) ) = ( ( ( ( B - X ) x. A ) + ( ( X - A ) x. B ) ) / ( B - A ) ) )
27	1, 4	subcld 10392	. . . . 5 |- ( ph -> ( B - X ) e. CC )
28	27, 2	mulcld 10060	. . . 4 |- ( ph -> ( ( B - X ) x. A ) e. CC )
29	4, 2	subcld 10392	. . . . 5 |- ( ph -> ( X - A ) e. CC )
30	29, 1	mulcld 10060	. . . 4 |- ( ph -> ( ( X - A ) x. B ) e. CC )
31	1, 2	subcld 10392	. . . 4 |- ( ph -> ( B - A ) e. CC )
32		bj-bary1.neq	. . . . . 6 |- ( ph -> A =/= B )
33	32	necomd 2849	. . . . 5 |- ( ph -> B =/= A )
34	1, 2, 33	subne0d 10401	. . . 4 |- ( ph -> ( B - A ) =/= 0 )
35	28, 30, 31, 34	divdird 10839	. . 3 |- ( ph -> ( ( ( ( B - X ) x. A ) + ( ( X - A ) x. B ) ) / ( B - A ) ) = ( ( ( ( B - X ) x. A ) / ( B - A ) ) + ( ( ( X - A ) x. B ) / ( B - A ) ) ) )
36	26, 35	eqtrd 2656	. 2 |- ( ph -> ( ( X x. ( B - A ) ) / ( B - A ) ) = ( ( ( ( B - X ) x. A ) / ( B - A ) ) + ( ( ( X - A ) x. B ) / ( B - A ) ) ) )
37	4, 31, 34	divcan4d 10807	. 2 |- ( ph -> ( ( X x. ( B - A ) ) / ( B - A ) ) = X )
38	27, 2, 31, 34	div23d 10838	. . 3 |- ( ph -> ( ( ( B - X ) x. A ) / ( B - A ) ) = ( ( ( B - X ) / ( B - A ) ) x. A ) )
39	29, 1, 31, 34	div23d 10838	. . 3 |- ( ph -> ( ( ( X - A ) x. B ) / ( B - A ) ) = ( ( ( X - A ) / ( B - A ) ) x. B ) )
40	38, 39	oveq12d 6668	. 2 |- ( ph -> ( ( ( ( B - X ) x. A ) / ( B - A ) ) + ( ( ( X - A ) x. B ) / ( B - A ) ) ) = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) )
41	36, 37, 40	3eqtr3d 2664	1 |- ( ph -> X = ( ( ( ( B - X ) / ( B - A ) ) x. A ) + ( ( ( X - A ) / ( B - A ) ) x. B ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794  (class class class)co 6650   CCcc 9934   0cc0 9936   + 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