Theorem lincmb01cmp 9025

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 108	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. ( 0 [,] 1 ) )
2		0re 7119	. . . . . . 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 7118	. . . . . . 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 8962	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
7	6	simp1bi 953	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) -> T e. RR )
8	7	adantl 271	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. RR )
9		difrp 8770	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
10	9	biimp3a 1276	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
11	10	adantr 270	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR+ )
12		eqid 2081	. . . . . . 7 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
13		eqid 2081	. . . . . . 7 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
14	12, 13	iccdil 9020	. . . . . 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 1170	. . . . 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 145	. . . 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 942	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. RR )
18		simpl1 941	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. RR )
19	17, 18	resubcld 7485	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
20	19	recnd 7147	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. CC )
21	20	mul02d 7496	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 x. ( B - A ) ) = 0 )
22	20	mulid2d 7137	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
23	21, 22	oveq12d 5550	. . . 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 2157	. . 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 7149	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) e. RR )
26		eqid 2081	. . . . 5 |- ( 0 + A ) = ( 0 + A )
27		eqid 2081	. . . . 5 |- ( ( B - A ) + A ) = ( ( B - A ) + A )
28	26, 27	iccshftr 9016	. . . 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 1170	. . 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 145	. 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 7147	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
32	17	recnd 7147	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
33	31, 32	mulcld 7139	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. CC )
34	18	recnd 7147	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
35	31, 34	mulcld 7139	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) e. CC )
36	33, 35, 34	subadd23d 7441	. . 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 7518	. . . 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 5547	. . 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 7372	. . . . . . . 8 |- ( ( 1 e. RR /\ T e. RR ) -> ( 1 - T ) e. RR )
40	4, 8, 39	sylancr 405	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. RR )
41	40, 18	remulcld 7149	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. RR )
42	41	recnd 7147	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. CC )
43	42, 33	addcomd 7259	. . . 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 7135	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
45	44, 31, 34	subdird 7519	. . . . . 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 7137	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
47	46	oveq1d 5547	. . . . . 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 2113	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) = ( A - ( T x. A ) ) )
49	48	oveq2d 5548	. . . 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 2113	. . 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 2123	. 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 7258	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 + A ) = A )
53	32, 34	npcand 7423	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( B - A ) + A ) = B )
54	52, 53	oveq12d 5550	. 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 2161	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 102   <-> wb 103   /\ w3a 919   e. wcel 1433   class class class wbr 3785  (class class class)co 5532   RRcr 6980   0cc0 6981   1c1 6982   + caddc 6984   x. cmul 6986   < clt 7153   <_ cle 7154   - cmin 7279   RR+crp 8734   [,]cicc 8914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-ltadd 7092  ax-pre-mulgt0 7093

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-po 4051  df-iso 4052  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-rp 8735  df-icc 8918

This theorem is referenced by:  iccf1o  9026