Theorem lincmb01cmp 10199

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 8146	. . . . . . 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 8145	. . . . . . 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 10135	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
7	6	simp1bi 1036	. . . . . . 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 9888	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A < B <-> ( B - A ) e. RR+ ) )
10	9	biimp3a 1379	. . . . . . 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 2229	. . . . . . 7 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
13		eqid 2229	. . . . . . 7 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
14	12, 13	iccdil 10194	. . . . . 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 1272	. . . . 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 1025	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. RR )
18		simpl1 1024	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. RR )
19	17, 18	resubcld 8527	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
20	19	recnd 8175	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. CC )
21	20	mul02d 8538	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 x. ( B - A ) ) = 0 )
22	20	mulid2d 8165	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
23	21, 22	oveq12d 6019	. . . 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 2308	. . 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 8177	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) e. RR )
26		eqid 2229	. . . . 5 |- ( 0 + A ) = ( 0 + A )
27		eqid 2229	. . . . 5 |- ( ( B - A ) + A ) = ( ( B - A ) + A )
28	26, 27	iccshftr 10190	. . . 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 1272	. . 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 8175	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
32	17	recnd 8175	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
33	31, 32	mulcld 8167	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. CC )
34	18	recnd 8175	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
35	31, 34	mulcld 8167	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) e. CC )
36	33, 35, 34	subadd23d 8479	. . 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 8560	. . . 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 6016	. . 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 8410	. . . . . . . 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 8177	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. RR )
42	41	recnd 8175	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. CC )
43	42, 33	addcomd 8297	. . . 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 8162	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
45	44, 31, 34	subdird 8561	. . . . . 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 8165	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
47	46	oveq1d 6016	. . . . . 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 2262	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) = ( A - ( T x. A ) ) )
49	48	oveq2d 6017	. . . 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 2262	. . 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 2272	. 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 8296	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 + A ) = A )
53	32, 34	npcand 8461	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( B - A ) + A ) = B )
54	52, 53	oveq12d 6019	. 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 2312	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 1002   e. wcel 2200   class class class wbr 4083  (class class class)co 6001   RRcr 7998   0cc0 7999   1c1 8000   + caddc 8002   x. cmul 8004   < clt 8181   <_ cle 8182   - cmin 8317   RR+crp 9849   [,]cicc 10087

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115  ax-pre-mulgt0 8116

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-rp 9850  df-icc 10091

This theorem is referenced by:  iccf1o  10200