Theorem lincmble 10337

Description: A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10336 but generalized to require merely A <_ B not A < B. (Contributed by Jim Kingdon, 13-May-2026.)

Assertion

Ref	Expression
lincmble	|- ( ( ( 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 lincmble

Step	Hyp	Ref	Expression
1		1red 8289	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. RR )
2		0re 8274	. . . . . . . 8 |- 0 e. RR
3		1re 8273	. . . . . . . 8 |- 1 e. RR
4	2, 3	elicc2i 10272	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
5	4	simp1bi 1039	. . . . . 6 |- ( T e. ( 0 [,] 1 ) -> T e. RR )
6	5	adantl 277	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> T e. RR )
7	1, 6	resubcld 8654	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. RR )
8		simpl1 1027	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A e. RR )
9	7, 8	remulcld 8304	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. RR )
10		simpl2 1028	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> B e. RR )
11	6, 10	remulcld 8304	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. RR )
12	9, 11	readdcld 8303	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. RR )
13		1cnd 8290	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
14	6	recnd 8302	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
15	13, 14	npcand 8588	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) + T ) = 1 )
16	15	oveq1d 6065	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. A ) = ( 1 x. A ) )
17	7	recnd 8302	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. CC )
18	8	recnd 8302	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
19	17, 14, 18	adddird 8299	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. A ) = ( ( ( 1 - T ) x. A ) + ( T x. A ) ) )
20	18	mullidd 8292	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
21	16, 19, 20	3eqtr3rd 2274	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A = ( ( ( 1 - T ) x. A ) + ( T x. A ) ) )
22	6, 8	remulcld 8304	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) e. RR )
23	4	simp2bi 1040	. . . . . 6 |- ( T e. ( 0 [,] 1 ) -> 0 <_ T )
24	23	adantl 277	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 0 <_ T )
25		simpl3 1029	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A <_ B )
26	8, 10, 6, 24, 25	lemul2ad 9214	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) <_ ( T x. B ) )
27	22, 11, 9, 26	leadd2dd 8834	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. A ) ) <_ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) )
28	21, 27	eqbrtrd 4131	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A <_ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) )
29	7, 10	remulcld 8304	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. B ) e. RR )
30	4	simp3bi 1041	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) -> T <_ 1 )
31		1red 8289	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) -> 1 e. RR )
32	31, 5	subge0d 8809	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) -> ( 0 <_ ( 1 - T ) <-> T <_ 1 ) )
33	30, 32	mpbird 167	. . . . . 6 |- ( T e. ( 0 [,] 1 ) -> 0 <_ ( 1 - T ) )
34	33	adantl 277	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 0 <_ ( 1 - T ) )
35	8, 10, 7, 34, 25	lemul2ad 9214	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) <_ ( ( 1 - T ) x. B ) )
36	9, 29, 11, 35	leadd1dd 8833	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) <_ ( ( ( 1 - T ) x. B ) + ( T x. B ) ) )
37	15	oveq1d 6065	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. B ) = ( 1 x. B ) )
38	10	recnd 8302	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
39	17, 14, 38	adddird 8299	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. B ) = ( ( ( 1 - T ) x. B ) + ( T x. B ) ) )
40	38	mullidd 8292	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. B ) = B )
41	37, 39, 40	3eqtr3d 2273	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. B ) + ( T x. B ) ) = B )
42	36, 41	breqtrd 4135	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) <_ B )
43		elicc2 10271	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) <-> ( ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. RR /\ A <_ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) /\ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) <_ B ) ) )
44	43	3adant3 1044	. . 3 |- ( ( A e. RR /\ B e. RR /\ A <_ B ) -> ( ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) <-> ( ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. RR /\ A <_ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) /\ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) <_ B ) ) )
45	44	adantr 276	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. ( A [,] B ) <-> ( ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. RR /\ A <_ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) /\ ( ( ( 1 - T ) x. A ) + ( T x. B ) ) <_ B ) ) )
46	12, 28, 42, 45	mpbir3and 1207	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 1005   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   <_ cle 8309   - cmin 8444   [,]cicc 10224

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-po 4417  df-iso 4418  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-icc 10228

This theorem is referenced by: (None)