Theorem affineequiv 24553

Description: Equivalence between two ways of expressing B as an affine combination of A and C. (Contributed by David Moews, 28-Feb-2017.)

Hypotheses

Ref	Expression
affineequiv.A	|- ( ph -> A e. CC )
affineequiv.B	|- ( ph -> B e. CC )
affineequiv.C	|- ( ph -> C e. CC )
affineequiv.D	|- ( ph -> D e. CC )

Assertion

Ref	Expression
affineequiv	|- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> ( C - B ) = ( D x. ( C - A ) ) ) )

Proof of Theorem affineequiv

Step	Hyp	Ref	Expression
1		affineequiv.C	. . . . . . . 8 |- ( ph -> C e. CC )
2		affineequiv.D	. . . . . . . . 9 |- ( ph -> D e. CC )
3	2, 1	mulcld 10060	. . . . . . . 8 |- ( ph -> ( D x. C ) e. CC )
4		affineequiv.A	. . . . . . . . 9 |- ( ph -> A e. CC )
5	2, 4	mulcld 10060	. . . . . . . 8 |- ( ph -> ( D x. A ) e. CC )
6	1, 3, 5	subsubd 10420	. . . . . . 7 |- ( ph -> ( C - ( ( D x. C ) - ( D x. A ) ) ) = ( ( C - ( D x. C ) ) + ( D x. A ) ) )
7	1, 3	subcld 10392	. . . . . . . 8 |- ( ph -> ( C - ( D x. C ) ) e. CC )
8	7, 5	addcomd 10238	. . . . . . 7 |- ( ph -> ( ( C - ( D x. C ) ) + ( D x. A ) ) = ( ( D x. A ) + ( C - ( D x. C ) ) ) )
9	6, 8	eqtr2d 2657	. . . . . 6 |- ( ph -> ( ( D x. A ) + ( C - ( D x. C ) ) ) = ( C - ( ( D x. C ) - ( D x. A ) ) ) )
10		1cnd 10056	. . . . . . . . 9 |- ( ph -> 1 e. CC )
11	10, 2, 1	subdird 10487	. . . . . . . 8 |- ( ph -> ( ( 1 - D ) x. C ) = ( ( 1 x. C ) - ( D x. C ) ) )
12	1	mulid2d 10058	. . . . . . . . 9 |- ( ph -> ( 1 x. C ) = C )
13	12	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( 1 x. C ) - ( D x. C ) ) = ( C - ( D x. C ) ) )
14	11, 13	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( 1 - D ) x. C ) = ( C - ( D x. C ) ) )
15	14	oveq2d 6666	. . . . . 6 |- ( ph -> ( ( D x. A ) + ( ( 1 - D ) x. C ) ) = ( ( D x. A ) + ( C - ( D x. C ) ) ) )
16		affineequiv.B	. . . . . . . 8 |- ( ph -> B e. CC )
17	1, 16	subcld 10392	. . . . . . . 8 |- ( ph -> ( C - B ) e. CC )
18	1, 4	subcld 10392	. . . . . . . . 9 |- ( ph -> ( C - A ) e. CC )
19	2, 18	mulcld 10060	. . . . . . . 8 |- ( ph -> ( D x. ( C - A ) ) e. CC )
20	16, 17, 19	addsubassd 10412	. . . . . . 7 |- ( ph -> ( ( B + ( C - B ) ) - ( D x. ( C - A ) ) ) = ( B + ( ( C - B ) - ( D x. ( C - A ) ) ) ) )
21	16, 1	pncan3d 10395	. . . . . . . 8 |- ( ph -> ( B + ( C - B ) ) = C )
22	2, 1, 4	subdid 10486	. . . . . . . 8 |- ( ph -> ( D x. ( C - A ) ) = ( ( D x. C ) - ( D x. A ) ) )
23	21, 22	oveq12d 6668	. . . . . . 7 |- ( ph -> ( ( B + ( C - B ) ) - ( D x. ( C - A ) ) ) = ( C - ( ( D x. C ) - ( D x. A ) ) ) )
24	20, 23	eqtr3d 2658	. . . . . 6 |- ( ph -> ( B + ( ( C - B ) - ( D x. ( C - A ) ) ) ) = ( C - ( ( D x. C ) - ( D x. A ) ) ) )
25	9, 15, 24	3eqtr4d 2666	. . . . 5 |- ( ph -> ( ( D x. A ) + ( ( 1 - D ) x. C ) ) = ( B + ( ( C - B ) - ( D x. ( C - A ) ) ) ) )
26	25	eqeq2d 2632	. . . 4 |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> B = ( B + ( ( C - B ) - ( D x. ( C - A ) ) ) ) ) )
27	16	addid1d 10236	. . . . 5 |- ( ph -> ( B + 0 ) = B )
28	27	eqeq1d 2624	. . . 4 |- ( ph -> ( ( B + 0 ) = ( B + ( ( C - B ) - ( D x. ( C - A ) ) ) ) <-> B = ( B + ( ( C - B ) - ( D x. ( C - A ) ) ) ) ) )
29		0cnd 10033	. . . . 5 |- ( ph -> 0 e. CC )
30	17, 19	subcld 10392	. . . . 5 |- ( ph -> ( ( C - B ) - ( D x. ( C - A ) ) ) e. CC )
31	16, 29, 30	addcand 10239	. . . 4 |- ( ph -> ( ( B + 0 ) = ( B + ( ( C - B ) - ( D x. ( C - A ) ) ) ) <-> 0 = ( ( C - B ) - ( D x. ( C - A ) ) ) ) )
32	26, 28, 31	3bitr2d 296	. . 3 |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> 0 = ( ( C - B ) - ( D x. ( C - A ) ) ) ) )
33		eqcom 2629	. . 3 |- ( 0 = ( ( C - B ) - ( D x. ( C - A ) ) ) <-> ( ( C - B ) - ( D x. ( C - A ) ) ) = 0 )
34	32, 33	syl6bb 276	. 2 |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> ( ( C - B ) - ( D x. ( C - A ) ) ) = 0 ) )
35	17, 19	subeq0ad 10402	. 2 |- ( ph -> ( ( ( C - B ) - ( D x. ( C - A ) ) ) = 0 <-> ( C - B ) = ( D x. ( C - A ) ) ) )
36	34, 35	bitrd 268	1 |- ( ph -> ( B = ( ( D x. A ) + ( ( 1 - D ) x. C ) ) <-> ( C - B ) = ( D x. ( C - A ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266

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

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-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-ltxr 10079  df-sub 10268

This theorem is referenced by:  affineequiv2  24554  angpieqvd  24558  chordthmlem2  24560  chordthmlem4  24562