Theorem lincmb01cmp 9907

Description: A linear combination of two reals which lies in the interval between them. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 8-Sep-2015.)

Assertion

Ref	Expression
lincmb01cmp	|- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) )

Proof of Theorem lincmb01cmp

Step	Hyp	Ref	Expression
1		simpr 109	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. ( 0 [,] 1 ) )
2		0re 7878	. . . . . . 7 |- 0 e. RR
3	2	a1i 9	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> 0 e. RR )
4		1re 7877	. . . . . . 7 |- 1 e. RR
5	4	a1i 9	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. RR )
6	2, 4	elicc2i 9843	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
7	6	simp1bi 997	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) -> T e. RR )
8	7	adantl 275	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. RR )
9		difrp 9599	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
10	9	biimp3a 1327	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
11	10	adantr 274	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR+ )
12		eqid 2157	. . . . . . 7 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
13		eqid 2157	. . . . . . 7 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
14	12, 13	iccdil 9902	. . . . . 6 |- ( ( ( 0 e. RR /\ 1 e. RR ) /\ ( T e. RR /\ ( B - A ) e. RR+ ) ) -> ( T e. ( 0 [,] 1 ) <-> ( T x. ( B - A ) ) e. ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) ) )
15	3, 5, 8, 11, 14	syl22anc 1221	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T e. ( 0 [,] 1 ) <-> ( T x. ( B - A ) ) e. ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) ) )
16	1, 15	mpbid 146	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) e. ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) )
17		simpl2 986	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. RR )
18		simpl1 985	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. RR )
19	17, 18	resubcld 8256	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
20	19	recnd 7906	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. CC )
21	20	mul02d 8267	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 x. ( B - A ) ) = 0 )
22	20	mulid2d 7896	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
23	21, 22	oveq12d 5842	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 0 x. ( B - A ) ) [,] ( 1 x. ( B - A ) ) ) = ( 0 [,] ( B - A ) ) )
24	16, 23	eleqtrd 2236	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) e. ( 0 [,] ( B - A ) ) )
25	8, 19	remulcld 7908	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) e. RR )
26		eqid 2157	. . . . 5 |- ( 0 + A ) = ( 0 + A )
27		eqid 2157	. . . . 5 |- ( ( B - A ) + A ) = ( ( B - A ) + A )
28	26, 27	iccshftr 9898	. . . 4 |- ( ( ( 0 e. RR /\ ( B - A ) e. RR ) /\ ( ( T x. ( B - A ) ) e. RR /\ A e. RR ) ) -> ( ( T x. ( B - A ) ) e. ( 0 [,] ( B - A ) ) <-> ( ( T x. ( B - A ) ) + A ) e. ( ( 0 + A ) [,] ( ( B - A ) + A ) ) ) )
29	3, 19, 25, 18, 28	syl22anc 1221	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( T x. ( B - A ) ) e. ( 0 [,] ( B - A ) ) <-> ( ( T x. ( B - A ) ) + A ) e. ( ( 0 + A ) [,] ( ( B - A ) + A ) ) ) )
30	24, 29	mpbid 146	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( T x. ( B - A ) ) + A ) e. ( ( 0 + A ) [,] ( ( B - A ) + A ) ) )
31	8	recnd 7906	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
32	17	recnd 7906	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
33	31, 32	mulcld 7898	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. CC )
34	18	recnd 7906	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
35	31, 34	mulcld 7898	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) e. CC )
36	33, 35, 34	subadd23d 8208	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( T x. B ) - ( T x. A ) ) + A ) = ( ( T x. B ) + ( A - ( T x. A ) ) ) )
37	31, 32, 34	subdid 8289	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) = ( ( T x. B ) - ( T x. A ) ) )
38	37	oveq1d 5839	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( T x. ( B - A ) ) + A ) = ( ( ( T x. B ) - ( T x. A ) ) + A ) )
39		resubcl 8139	. . . . . . . 8 |- ( ( 1 e. RR /\ T e. RR ) -> ( 1 - T ) e. RR )
40	4, 8, 39	sylancr 411	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. RR )
41	40, 18	remulcld 7908	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. RR )
42	41	recnd 7906	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. CC )
43	42, 33	addcomd 8026	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) = ( ( T x. B ) + ( ( 1 - T ) x. A ) ) )
44		1cnd 7894	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
45	44, 31, 34	subdird 8290	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) = ( ( 1 x. A ) - ( T x. A ) ) )
46	34	mulid2d 7896	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
47	46	oveq1d 5839	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 x. A ) - ( T x. A ) ) = ( A - ( T x. A ) ) )
48	45, 47	eqtrd 2190	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) = ( A - ( T x. A ) ) )
49	48	oveq2d 5840	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( T x. B ) + ( ( 1 - T ) x. A ) ) = ( ( T x. B ) + ( A - ( T x. A ) ) ) )
50	43, 49	eqtrd 2190	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) = ( ( T x. B ) + ( A - ( T x. A ) ) ) )
51	36, 38, 50	3eqtr4d 2200	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( T x. ( B - A ) ) + A ) = ( ( ( 1 - T ) x. A ) + ( T x. B ) ) )
52	34	addid2d 8025	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 + A ) = A )
53	32, 34	npcand 8190	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( B - A ) + A ) = B )
54	52, 53	oveq12d 5842	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 0 + A ) [,] ( ( B - A ) + A ) ) = ( A [,] B ) )
55	30, 51, 54	3eltr3d 2240	1 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   e. wcel 2128   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733   + caddc 7735   x. cmul 7737   < clt 7912   <_ cle 7913   - cmin 8046   RR+crp 9560   [,]cicc 9795

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848  ax-pre-mulgt0 7849

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-rp 9561  df-icc 9799

This theorem is referenced by:  iccf1o  9908