Theorem lincmble 10283

Description: A linear combination of two reals which lies in the interval between them. Like lincmb01cmp 10282 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 8237	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. RR )
2		0re 8222	. . . . . . . 8 |- 0 e. RR
3		1re 8221	. . . . . . . 8 |- 1 e. RR
4	2, 3	elicc2i 10218	. . . . . . 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 8602	. . . 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 8252	. . 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 8252	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( T x. B ) e. RR )
12	9, 11	readdcld 8251	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) e. RR )
13		1cnd 8238	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> 1 e. CC )
14	6	recnd 8250	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> T e. CC )
15	13, 14	npcand 8536	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( 1 - T ) + T ) = 1 )
16	15	oveq1d 6043	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. A ) = ( 1 x. A ) )
17	7	recnd 8250	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 - T ) e. CC )
18	8	recnd 8250	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> A e. CC )
19	17, 14, 18	adddird 8247	. . . 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 8240	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. A ) = A )
21	16, 19, 20	3eqtr3rd 2273	. . 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 8252	. . . 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 9162	. . . 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 8782	. . 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 4115	. 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 8252	. . . 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 8237	. . . . . . . 8 |- ( T e. ( 0 [,] 1 ) -> 1 e. RR )
32	31, 5	subge0d 8757	. . . . . . 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 9162	. . . 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 8781	. . 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 6043	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) + T ) x. B ) = ( 1 x. B ) )
38	10	recnd 8250	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> B e. CC )
39	17, 14, 38	adddird 8247	. . . 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 8240	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( 1 x. B ) = B )
41	37, 39, 40	3eqtr3d 2272	. . 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 4119	. 2 |- ( ( ( A e. RR /\ B e. RR /\ A <_ B ) /\ T e. ( 0 [,] 1 ) ) -> ( ( ( 1 - T ) x. A ) + ( T x. B ) ) <_ B )
43		elicc2 10217	. . . 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 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8074   0cc0 8075   1c1 8076   + caddc 8078   x. cmul 8080   <_ cle 8257   - cmin 8392   [,]cicc 10170

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-icc 10174

This theorem is referenced by: (None)