Theorem lincmb01icc 15879

Description: A linear combination of two reals which lies in the interval between them.

Assertion

Ref	Expression
lincmb01icc	|- ((A e. RR /\ B e. RR) -> ((C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) e. (A[,]B)))

Proof of Theorem lincmb01icc

Step	Hyp	Ref	Expression
1		iccssre 7565	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A[,]B) C_ RR)
2	1	sseld 2619	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (C e. (A[,]B) -> C e. RR))
3	2	imp 377	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ C e. (A[,]B)) -> C e. RR)
4	3	adantrr 431	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B))) -> C e. RR)
5	1	sseld 2619	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (D e. (A[,]B) -> D e. RR))
6	5	imp 377	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ D e. (A[,]B)) -> D e. RR)
7	6	adantrl 430	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B))) -> D e. RR)
8		lttri4 6684	. . . . 5 |- ((C e. RR /\ D e. RR) -> (C < D \/ C = D \/ D < C))
9	4, 7, 8	syl11anc 524	. . . 4 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B))) -> (C < D \/ C = D \/ D < C))
10	9	3adantr3 1037	. . 3 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> (C < D \/ C = D \/ D < C))
11		iccss2 15856	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> ((C e. (A[,]B) /\ D e. (A[,]B)) -> (C[,]D) C_ (A[,]B)))
12	11	imp 377	. . . . . . 7 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B))) -> (C[,]D) C_ (A[,]B))
13	12	3adantr3 1037	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> (C[,]D) C_ (A[,]B))
14	13	adantr 425	. . . . 5 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ C < D) -> (C[,]D) C_ (A[,]B))
15		lincmb01cmp 15878	. . . . . . . . . 10 |- (((C e. RR /\ D e. RR /\ C < D) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) e. (C[,]D))
16	15	ex 402	. . . . . . . . 9 |- ((C e. RR /\ D e. RR /\ C < D) -> (T e. (0[,]1) -> (((1 - T) x. C) + (T x. D)) e. (C[,]D)))
17	16	3expa 1067	. . . . . . . 8 |- (((C e. RR /\ D e. RR) /\ C < D) -> (T e. (0[,]1) -> (((1 - T) x. C) + (T x. D)) e. (C[,]D)))
18	17	imp 377	. . . . . . 7 |- ((((C e. RR /\ D e. RR) /\ C < D) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) e. (C[,]D))
19	18	an1rs 547	. . . . . 6 |- ((((C e. RR /\ D e. RR) /\ T e. (0[,]1)) /\ C < D) -> (((1 - T) x. C) + (T x. D)) e. (C[,]D))
20	4, 7	jca 310	. . . . . . . 8 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B))) -> (C e. RR /\ D e. RR))
21	20	3adantr3 1037	. . . . . . 7 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> (C e. RR /\ D e. RR))
22		simpr3 884	. . . . . . 7 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> T e. (0[,]1))
23	21, 22	jca 310	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> ((C e. RR /\ D e. RR) /\ T e. (0[,]1)))
24	19, 23	sylan 497	. . . . 5 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ C < D) -> (((1 - T) x. C) + (T x. D)) e. (C[,]D))
25	14, 24	sseldd 2620	. . . 4 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ C < D) -> (((1 - T) x. C) + (T x. D)) e. (A[,]B))
26		opreq2 4890	. . . . . . 7 |- (C = D -> ((1 - T) x. C) = ((1 - T) x. D))
27	26	opreq1d 4897	. . . . . 6 |- (C = D -> (((1 - T) x. C) + (T x. D)) = (((1 - T) x. D) + (T x. D)))
28		0re 6603	. . . . . . . . . . . 12 |- 0 e. RR
29		1re 6598	. . . . . . . . . . . 12 |- 1 e. RR
30		iccssre 7565	. . . . . . . . . . . 12 |- ((0 e. RR /\ 1 e. RR) -> (0[,]1) C_ RR)
31	28, 29, 30	mp2an 761	. . . . . . . . . . 11 |- (0[,]1) C_ RR
32	31	sseli 2617	. . . . . . . . . 10 |- (T e. (0[,]1) -> T e. RR)
33	32	recnd 6468	. . . . . . . . 9 |- (T e. (0[,]1) -> T e. CC)
34	33	ad2antll 443	. . . . . . . 8 |- (((A e. RR /\ B e. RR) /\ (D e. (A[,]B) /\ T e. (0[,]1))) -> T e. CC)
35	6	recnd 6468	. . . . . . . . 9 |- (((A e. RR /\ B e. RR) /\ D e. (A[,]B)) -> D e. CC)
36	35	adantrr 431	. . . . . . . 8 |- (((A e. RR /\ B e. RR) /\ (D e. (A[,]B) /\ T e. (0[,]1))) -> D e. CC)
37		ax1cn 6422	. . . . . . . . . . . 12 |- 1 e. CC
38		npcan 6559	. . . . . . . . . . . 12 |- ((1 e. CC /\ T e. CC) -> ((1 - T) + T) = 1)
39	37, 38	mpan 759	. . . . . . . . . . 11 |- (T e. CC -> ((1 - T) + T) = 1)
40	39	adantr 425	. . . . . . . . . 10 |- ((T e. CC /\ D e. CC) -> ((1 - T) + T) = 1)
41	40	opreq1d 4897	. . . . . . . . 9 |- ((T e. CC /\ D e. CC) -> (((1 - T) + T) x. D) = (1 x. D))
42		adddir 6472	. . . . . . . . . . 11 |- (((1 - T) e. CC /\ T e. CC /\ D e. CC) -> (((1 - T) + T) x. D) = (((1 - T) x. D) + (T x. D)))
43	42	3expa 1067	. . . . . . . . . 10 |- ((((1 - T) e. CC /\ T e. CC) /\ D e. CC) -> (((1 - T) + T) x. D) = (((1 - T) x. D) + (T x. D)))
44		subcl 6524	. . . . . . . . . . . 12 |- ((1 e. CC /\ T e. CC) -> (1 - T) e. CC)
45	37, 44	mpan 759	. . . . . . . . . . 11 |- (T e. CC -> (1 - T) e. CC)
46	45	ancri 321	. . . . . . . . . 10 |- (T e. CC -> ((1 - T) e. CC /\ T e. CC))
47	43, 46	sylan 497	. . . . . . . . 9 |- ((T e. CC /\ D e. CC) -> (((1 - T) + T) x. D) = (((1 - T) x. D) + (T x. D)))
48		mulid2 6578	. . . . . . . . . 10 |- (D e. CC -> (1 x. D) = D)
49	48	adantl 424	. . . . . . . . 9 |- ((T e. CC /\ D e. CC) -> (1 x. D) = D)
50	41, 47, 49	3eqtr3d 1934	. . . . . . . 8 |- ((T e. CC /\ D e. CC) -> (((1 - T) x. D) + (T x. D)) = D)
51	34, 36, 50	syl11anc 524	. . . . . . 7 |- (((A e. RR /\ B e. RR) /\ (D e. (A[,]B) /\ T e. (0[,]1))) -> (((1 - T) x. D) + (T x. D)) = D)
52	51	3adantr1 1035	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> (((1 - T) x. D) + (T x. D)) = D)
53	27, 52	sylan9eqr 1951	. . . . 5 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ C = D) -> (((1 - T) x. C) + (T x. D)) = D)
54		simplr2 919	. . . . 5 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ C = D) -> D e. (A[,]B))
55	53, 54	eqeltrd 1971	. . . 4 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ C = D) -> (((1 - T) x. C) + (T x. D)) e. (A[,]B))
56		iccss2 15856	. . . . . . . . 9 |- ((A e. RR /\ B e. RR) -> ((D e. (A[,]B) /\ C e. (A[,]B)) -> (D[,]C) C_ (A[,]B)))
57	56	imp 377	. . . . . . . 8 |- (((A e. RR /\ B e. RR) /\ (D e. (A[,]B) /\ C e. (A[,]B))) -> (D[,]C) C_ (A[,]B))
58	57	ancom2s 545	. . . . . . 7 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B))) -> (D[,]C) C_ (A[,]B))
59	58	3adantr3 1037	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> (D[,]C) C_ (A[,]B))
60	59	adantr 425	. . . . 5 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ D < C) -> (D[,]C) C_ (A[,]B))
61		mulcl 6456	. . . . . . . . . . . . . . . . 17 |- (((1 - T) e. CC /\ C e. CC) -> ((1 - T) x. C) e. CC)
62		iirev 15871	. . . . . . . . . . . . . . . . . . 19 |- (T e. (0[,]1) -> (1 - T) e. (0[,]1))
63	31	sseli 2617	. . . . . . . . . . . . . . . . . . 19 |- ((1 - T) e. (0[,]1) -> (1 - T) e. RR)
64	62, 63	syl 12	. . . . . . . . . . . . . . . . . 18 |- (T e. (0[,]1) -> (1 - T) e. RR)
65	64	recnd 6468	. . . . . . . . . . . . . . . . 17 |- (T e. (0[,]1) -> (1 - T) e. CC)
66		recn 6466	. . . . . . . . . . . . . . . . 17 |- (C e. RR -> C e. CC)
67	61, 65, 66	syl2an 503	. . . . . . . . . . . . . . . 16 |- ((T e. (0[,]1) /\ C e. RR) -> ((1 - T) x. C) e. CC)
68	67	ancoms 484	. . . . . . . . . . . . . . 15 |- ((C e. RR /\ T e. (0[,]1)) -> ((1 - T) x. C) e. CC)
69	68	adantll 428	. . . . . . . . . . . . . 14 |- (((D e. RR /\ C e. RR) /\ T e. (0[,]1)) -> ((1 - T) x. C) e. CC)
70		mulcl 6456	. . . . . . . . . . . . . . . . 17 |- ((T e. CC /\ D e. CC) -> (T x. D) e. CC)
71		recn 6466	. . . . . . . . . . . . . . . . 17 |- (D e. RR -> D e. CC)
72	70, 33, 71	syl2an 503	. . . . . . . . . . . . . . . 16 |- ((T e. (0[,]1) /\ D e. RR) -> (T x. D) e. CC)
73	72	ancoms 484	. . . . . . . . . . . . . . 15 |- ((D e. RR /\ T e. (0[,]1)) -> (T x. D) e. CC)
74	73	adantlr 429	. . . . . . . . . . . . . 14 |- (((D e. RR /\ C e. RR) /\ T e. (0[,]1)) -> (T x. D) e. CC)
75		addcom 6458	. . . . . . . . . . . . . 14 |- ((((1 - T) x. C) e. CC /\ (T x. D) e. CC) -> (((1 - T) x. C) + (T x. D)) = ((T x. D) + ((1 - T) x. C)))
76	69, 74, 75	syl11anc 524	. . . . . . . . . . . . 13 |- (((D e. RR /\ C e. RR) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) = ((T x. D) + ((1 - T) x. C)))
77	76	3adantl3 1034	. . . . . . . . . . . 12 |- (((D e. RR /\ C e. RR /\ D < C) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) = ((T x. D) + ((1 - T) x. C)))
78		nncan 6635	. . . . . . . . . . . . . . . . . 18 |- ((1 e. CC /\ T e. CC) -> (1 - (1 - T)) = T)
79	37, 78	mpan 759	. . . . . . . . . . . . . . . . 17 |- (T e. CC -> (1 - (1 - T)) = T)
80	79	eqcomd 1889	. . . . . . . . . . . . . . . 16 |- (T e. CC -> T = (1 - (1 - T)))
81	80	opreq1d 4897	. . . . . . . . . . . . . . 15 |- (T e. CC -> (T x. D) = ((1 - (1 - T)) x. D))
82	81	opreq1d 4897	. . . . . . . . . . . . . 14 |- (T e. CC -> ((T x. D) + ((1 - T) x. C)) = (((1 - (1 - T)) x. D) + ((1 - T) x. C)))
83	33, 82	syl 12	. . . . . . . . . . . . 13 |- (T e. (0[,]1) -> ((T x. D) + ((1 - T) x. C)) = (((1 - (1 - T)) x. D) + ((1 - T) x. C)))
84	83	adantl 424	. . . . . . . . . . . 12 |- (((D e. RR /\ C e. RR /\ D < C) /\ T e. (0[,]1)) -> ((T x. D) + ((1 - T) x. C)) = (((1 - (1 - T)) x. D) + ((1 - T) x. C)))
85	77, 84	eqtrd 1925	. . . . . . . . . . 11 |- (((D e. RR /\ C e. RR /\ D < C) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) = (((1 - (1 - T)) x. D) + ((1 - T) x. C)))
86		lincmb01cmp 15878	. . . . . . . . . . . 12 |- (((D e. RR /\ C e. RR /\ D < C) /\ (1 - T) e. (0[,]1)) -> (((1 - (1 - T)) x. D) + ((1 - T) x. C)) e. (D[,]C))
87	86, 62	sylan2 500	. . . . . . . . . . 11 |- (((D e. RR /\ C e. RR /\ D < C) /\ T e. (0[,]1)) -> (((1 - (1 - T)) x. D) + ((1 - T) x. C)) e. (D[,]C))
88	85, 87	eqeltrd 1971	. . . . . . . . . 10 |- (((D e. RR /\ C e. RR /\ D < C) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) e. (D[,]C))
89	88	ex 402	. . . . . . . . 9 |- ((D e. RR /\ C e. RR /\ D < C) -> (T e. (0[,]1) -> (((1 - T) x. C) + (T x. D)) e. (D[,]C)))
90	89	3expa 1067	. . . . . . . 8 |- (((D e. RR /\ C e. RR) /\ D < C) -> (T e. (0[,]1) -> (((1 - T) x. C) + (T x. D)) e. (D[,]C)))
91	90	imp 377	. . . . . . 7 |- ((((D e. RR /\ C e. RR) /\ D < C) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) e. (D[,]C))
92	91	an1rs 547	. . . . . 6 |- ((((D e. RR /\ C e. RR) /\ T e. (0[,]1)) /\ D < C) -> (((1 - T) x. C) + (T x. D)) e. (D[,]C))
93	7, 4	jca 310	. . . . . . . 8 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B))) -> (D e. RR /\ C e. RR))
94	93	3adantr3 1037	. . . . . . 7 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> (D e. RR /\ C e. RR))
95	94, 22	jca 310	. . . . . 6 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> ((D e. RR /\ C e. RR) /\ T e. (0[,]1)))
96	92, 95	sylan 497	. . . . 5 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ D < C) -> (((1 - T) x. C) + (T x. D)) e. (D[,]C))
97	60, 96	sseldd 2620	. . . 4 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ D < C) -> (((1 - T) x. C) + (T x. D)) e. (A[,]B))
98	25, 55, 97	3jaodan 1163	. . 3 |- ((((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) /\ (C < D \/ C = D \/ D < C)) -> (((1 - T) x. C) + (T x. D)) e. (A[,]B))
99	10, 98	mpdan 768	. 2 |- (((A e. RR /\ B e. RR) /\ (C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1))) -> (((1 - T) x. C) + (T x. D)) e. (A[,]B))
100	99	ex 402	1 |- ((A e. RR /\ B e. RR) -> ((C e. (A[,]B) /\ D e. (A[,]B) /\ T e. (0[,]1)) -> (((1 - T) x. C) + (T x. D)) e. (A[,]B)))

Syntax hints:   -> wi 3   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300   C_ wss 2593   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   < clt 6653  [,]cicc 7527

This theorem is referenced by:  reparphtlem2 16064  reparpht 16065  pcorevlem 16086

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-rp 7232  df-icc 7531

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