Theorem lincmble 10385

Description: A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10384 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 8331	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. RR )
2		0re 8316	. . . . . . . 8 |- 0 e. RR
3		1re 8315	. . . . . . . 8 |- 1 e. RR
4	2, 3	elicc2i 10320	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) <-> ( T e. RR /\ 0 <_ T /\ T <_ 1 ) )
5	4	simp1bi 1043	. . . . . 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 8698	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. RR )
8		simpl1 1031	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A e. RR )
9	7, 8	remulcld 8346	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. A ) e. RR )
10		simpl2 1032	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> B e. RR )
11	6, 10	remulcld 8346	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. RR )
12	9, 11	readdcld 8345	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. RR )
13		1cnd 8332	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
14	6	recnd 8344	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
15	13, 14	npcand 8631	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) + T ) = 1 )
16	15	oveq1d 6090	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. A ) = ( 1 x. A ) )
17	7	recnd 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. CC )
18	8	recnd 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
19	17, 14, 18	adddird 8341	. . . 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 8334	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
21	16, 19, 20	3eqtr3rd 2280	. . 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 8346	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. A ) e. RR )
23	4	simp2bi 1044	. . . . . 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 1033	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A <_ B )
26	8, 10, 6, 24, 25	lemul2ad 9260	. . . 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 8878	. . 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 4147	. 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 8346	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) x. B ) e. RR )
30	4	simp3bi 1045	. . . . . . 7 |- ( T e. ( 0 [,] 1 ) -> T <_ 1 )
31		1red 8331	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) -> 1 e. RR )
32	31, 5	subge0d 8853	. . . . . . 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 9260	. . . 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 8877	. . 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 6090	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. B ) = ( 1 x. B ) )
38	10	recnd 8344	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
39	17, 14, 38	adddird 8341	. . . 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 8334	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. B ) = B )
41	37, 39, 40	3eqtr3d 2279	. . 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 4151	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) <_ B )
43		elicc2 10319	. . . 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 1048	. . 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 1211	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 1009   e. wcel 2209   class class class wbr 4125  (class class class)co 6075   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   <_ cle 8351   - cmin 8487   [,]cicc 10272

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-icc 10276

This theorem is referenced by: (None)