Theorem lincmb01cmp 10160

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 110	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. ( 0 [,] 1 ) )
2		0re 8107	. . . . . . 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 8106	. . . . . . 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 10096	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
7	6	simp1bi 1015	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) -> T e. RR )
8	7	adantl 277	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. RR )
9		difrp 9849	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
10	9	biimp3a 1358	. . . . . . 7 |- ( ( A e. RR /\ B e. RR /\ A < B ) -> ( B - A ) e. RR+ )
11	10	adantr 276	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR+ )
12		eqid 2207	. . . . . . 7 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
13		eqid 2207	. . . . . . 7 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
14	12, 13	iccdil 10155	. . . . . 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 1251	. . . . 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 147	. . . 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 1004	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. RR )
18		simpl1 1003	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. RR )
19	17, 18	resubcld 8488	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
20	19	recnd 8136	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. CC )
21	20	mul02d 8499	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 x. ( B - A ) ) = 0 )
22	20	mulid2d 8126	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
23	21, 22	oveq12d 5985	. . . 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 2286	. . 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 8138	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) e. RR )
26		eqid 2207	. . . . 5 |- ( 0 + A ) = ( 0 + A )
27		eqid 2207	. . . . 5 |- ( ( B - A ) + A ) = ( ( B - A ) + A )
28	26, 27	iccshftr 10151	. . . 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 1251	. . 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 147	. 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 8136	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
32	17	recnd 8136	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
33	31, 32	mulcld 8128	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. CC )
34	18	recnd 8136	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
35	31, 34	mulcld 8128	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) e. CC )
36	33, 35, 34	subadd23d 8440	. . 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 8521	. . . 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 5982	. . 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 8371	. . . . . . . 8 |- ( ( 1 e. RR /\ T e. RR ) -> ( 1 - T ) e. RR )
40	4, 8, 39	sylancr 414	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. RR )
41	40, 18	remulcld 8138	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. RR )
42	41	recnd 8136	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. CC )
43	42, 33	addcomd 8258	. . . 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 8123	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
45	44, 31, 34	subdird 8522	. . . . . 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 8126	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
47	46	oveq1d 5982	. . . . . 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 2240	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) = ( A - ( T x. A ) ) )
49	48	oveq2d 5983	. . . 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 2240	. . 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 2250	. 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	addlidd 8257	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 + A ) = A )
53	32, 34	npcand 8422	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( B - A ) + A ) = B )
54	52, 53	oveq12d 5985	. 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 2290	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 104   <-> wb 105   /\ w3a 981   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   RRcr 7959   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   < clt 8142   <_ cle 8143   - cmin 8278   RR+crp 9810   [,]cicc 10048

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-ltadd 8076  ax-pre-mulgt0 8077

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-rp 9811  df-icc 10052

This theorem is referenced by:  iccf1o  10161