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Theorem lincmb01cmp 9907
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
StepHypRef Expression
1 simpr 109 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
2 0re 7878 . . . . . . 7  |-  0  e.  RR
32a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  0  e.  RR )
4 1re 7877 . . . . . . 7  |-  1  e.  RR
54a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  RR )
62, 4elicc2i 9843 . . . . . . . 8  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
76simp1bi 997 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
87adantl 275 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  RR )
9 difrp 9599 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
109biimp3a 1327 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
1110adantr 274 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR+ )
12 eqid 2157 . . . . . . 7  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
13 eqid 2157 . . . . . . 7  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
1412, 13iccdil 9902 . . . . . 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 ) ) ) ) )
153, 5, 8, 11, 14syl22anc 1221 . . . . 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 ) ) ) ) )
161, 15mpbid 146 . . . 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 986 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
18 simpl1 985 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1917, 18resubcld 8256 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
2019recnd 7906 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  CC )
2120mul02d 8267 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  x.  ( B  -  A
) )  =  0 )
2220mulid2d 7896 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  ( B  -  A
) )  =  ( B  -  A ) )
2321, 22oveq12d 5842 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 0  x.  ( B  -  A ) ) [,] ( 1  x.  ( B  -  A )
) )  =  ( 0 [,] ( B  -  A ) ) )
2416, 23eleqtrd 2236 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  ( 0 [,] ( B  -  A ) ) )
258, 19remulcld 7908 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  RR )
26 eqid 2157 . . . . 5  |-  ( 0  +  A )  =  ( 0  +  A
)
27 eqid 2157 . . . . 5  |-  ( ( B  -  A )  +  A )  =  ( ( B  -  A )  +  A
)
2826, 27iccshftr 9898 . . . 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 ) ) ) )
293, 19, 25, 18, 28syl22anc 1221 . . 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 ) ) ) )
3024, 29mpbid 146 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( T  x.  ( B  -  A ) )  +  A )  e.  ( ( 0  +  A
) [,] ( ( B  -  A )  +  A ) ) )
318recnd 7906 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  CC )
3217recnd 7906 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
3331, 32mulcld 7898 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  B )  e.  CC )
3418recnd 7906 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
3531, 34mulcld 7898 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  A )  e.  CC )
3633, 35, 34subadd23d 8208 . . 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 ) ) ) )
3731, 32, 34subdid 8289 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  =  ( ( T  x.  B
)  -  ( T  x.  A ) ) )
3837oveq1d 5839 . . 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 8139 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
404, 8, 39sylancr 411 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  -  T )  e.  RR )
4140, 18remulcld 7908 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  RR )
4241recnd 7906 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  CC )
4342, 33addcomd 8026 . . . 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 7894 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
4544, 31, 34subdird 8290 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( ( 1  x.  A
)  -  ( T  x.  A ) ) )
4634mulid2d 7896 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
4746oveq1d 5839 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  x.  A )  -  ( T  x.  A
) )  =  ( A  -  ( T  x.  A ) ) )
4845, 47eqtrd 2190 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( A  -  ( T  x.  A ) ) )
4948oveq2d 5840 . . . 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 ) ) ) )
5043, 49eqtrd 2190 . . 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 ) ) ) )
5136, 38, 503eqtr4d 2200 . 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 ) ) )
5234addid2d 8025 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  +  A )  =  A )
5332, 34npcand 8190 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( B  -  A )  +  A )  =  B )
5452, 53oveq12d 5842 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 0  +  A ) [,] ( ( B  -  A )  +  A
) )  =  ( A [,] B ) )
5530, 51, 543eltr3d 2240 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  A )  +  ( T  x.  B
) )  e.  ( A [,] B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    e. wcel 2128   class class class wbr 3965  (class class class)co 5824   RRcr 7731   0cc0 7732   1c1 7733    + caddc 7735    x. cmul 7737    < clt 7912    <_ cle 7913    - cmin 8046   RR+crp 9560   [,]cicc 9795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-ltadd 7848  ax-pre-mulgt0 7849
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-rp 9561  df-icc 9799
This theorem is referenced by:  iccf1o  9908
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