Theorem lincmb01cmp 10125

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 8072	. . . . . . 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 8071	. . . . . . 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 10061	. . . . . . . 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 9814	. . . . . . . 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 2205	. . . . . . 7 |- ( 0 x. ( B - A ) ) = ( 0 x. ( B - A ) )
13		eqid 2205	. . . . . . 7 |- ( 1 x. ( B - A ) ) = ( 1 x. ( B - A ) )
14	12, 13	iccdil 10120	. . . . . 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 8453	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. RR )
20	19	recnd 8101	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( B - A ) e. CC )
21	20	mul02d 8464	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 x. ( B - A ) ) = 0 )
22	20	mulid2d 8091	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. ( B - A ) ) = ( B - A ) )
23	21, 22	oveq12d 5962	. . . 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 2284	. . 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 8103	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. ( B - A ) ) e. RR )
26		eqid 2205	. . . . 5 |- ( 0 + A ) = ( 0 + A )
27		eqid 2205	. . . . 5 |- ( ( B - A ) + A ) = ( ( B - A ) + A )
28	26, 27	iccshftr 10116	. . . 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 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
32	17	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
33	31, 32	mulcld 8093	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. CC )
34	18	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
35	31, 34	mulcld 8093	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) e. CC )
36	33, 35, 34	subadd23d 8405	. . 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 8486	. . . 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 5959	. . 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 8336	. . . . . . . 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 8103	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. RR )
42	41	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. CC )
43	42, 33	addcomd 8223	. . . 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 8088	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
45	44, 31, 34	subdird 8487	. . . . . 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 8091	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
47	46	oveq1d 5959	. . . . . 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 2238	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) = ( A - ( T x. A ) ) )
49	48	oveq2d 5960	. . . 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 2238	. . 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 2248	. 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 8222	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( 0 + A ) = A )
53	32, 34	npcand 8387	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A < B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( B - A ) + A ) = B )
54	52, 53	oveq12d 5962	. 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 2288	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 2176   class class class wbr 4044  (class class class)co 5944   RRcr 7924   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   < clt 8107   <_ cle 8108   - cmin 8243   RR+crp 9775   [,]cicc 10013

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-ltadd 8041  ax-pre-mulgt0 8042

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-id 4340  df-po 4343  df-iso 4344  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-rp 9776  df-icc 10017

This theorem is referenced by:  iccf1o  10126