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Theorem lincmb01cmp 8838
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 103 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
2 0re 7008 . . . . . . 7  |-  0  e.  RR
32a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  0  e.  RR )
4 1re 7007 . . . . . . 7  |-  1  e.  RR
54a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  RR )
62, 4elicc2i 8775 . . . . . . . 8  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
76simp1bi 919 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
87adantl 262 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  RR )
9 difrp 8590 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
109biimp3a 1235 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
1110adantr 261 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR+ )
12 eqid 2040 . . . . . . 7  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
13 eqid 2040 . . . . . . 7  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
1412, 13iccdil 8833 . . . . . 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 1136 . . . . 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 135 . . . 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 908 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
18 simpl1 907 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1917, 18resubcld 7358 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
2019recnd 7034 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  CC )
2120mul02d 7368 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  x.  ( B  -  A
) )  =  0 )
2220mulid2d 7026 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  ( B  -  A
) )  =  ( B  -  A ) )
2321, 22oveq12d 5517 . . . 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 2116 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  ( 0 [,] ( B  -  A ) ) )
258, 19remulcld 7036 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  RR )
26 eqid 2040 . . . . 5  |-  ( 0  +  A )  =  ( 0  +  A
)
27 eqid 2040 . . . . 5  |-  ( ( B  -  A )  +  A )  =  ( ( B  -  A )  +  A
)
2826, 27iccshftr 8829 . . . 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 1136 . . 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 135 . 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 7034 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  CC )
3217recnd 7034 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
3331, 32mulcld 7028 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  B )  e.  CC )
3418recnd 7034 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
3531, 34mulcld 7028 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  A )  e.  CC )
3633, 35, 34subadd23d 7323 . . 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 7390 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  =  ( ( T  x.  B
)  -  ( T  x.  A ) ) )
3837oveq1d 5514 . . 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 7254 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
404, 8, 39sylancr 393 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  -  T )  e.  RR )
4140, 18remulcld 7036 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  RR )
4241recnd 7034 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  CC )
4342, 33addcomd 7144 . . . 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 7024 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
4544, 31, 34subdird 7391 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( ( 1  x.  A
)  -  ( T  x.  A ) ) )
4634mulid2d 7026 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
4746oveq1d 5514 . . . . . 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 2072 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( A  -  ( T  x.  A ) ) )
4948oveq2d 5515 . . . 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 2072 . . 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 2082 . 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 7143 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  +  A )  =  A )
5332, 34npcand 7305 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( B  -  A )  +  A )  =  B )
5452, 53oveq12d 5517 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 0  +  A ) [,] ( ( B  -  A )  +  A
) )  =  ( A [,] B ) )
5530, 51, 543eltr3d 2120 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 97    <-> wb 98    /\ w3a 885    e. wcel 1393   class class class wbr 3761  (class class class)co 5499   RRcr 6869   0cc0 6870   1c1 6871    + caddc 6873    x. cmul 6875    < clt 7040    <_ cle 7041    - cmin 7162   RR+crp 8554   [,]cicc 8727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941  ax-un 4166  ax-setind 4256  ax-cnex 6956  ax-resscn 6957  ax-1cn 6958  ax-1re 6959  ax-icn 6960  ax-addcl 6961  ax-addrcl 6962  ax-mulcl 6963  ax-mulrcl 6964  ax-addcom 6965  ax-mulcom 6966  ax-addass 6967  ax-mulass 6968  ax-distr 6969  ax-i2m1 6970  ax-1rid 6972  ax-0id 6973  ax-rnegex 6974  ax-precex 6975  ax-cnre 6976  ax-pre-ltirr 6977  ax-pre-ltwlin 6978  ax-pre-lttrn 6979  ax-pre-ltadd 6981  ax-pre-mulgt0 6982
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-br 3762  df-opab 3816  df-id 4027  df-po 4030  df-iso 4031  df-xp 4338  df-rel 4339  df-cnv 4340  df-co 4341  df-dm 4342  df-iota 4854  df-fun 4891  df-fv 4897  df-riota 5455  df-ov 5502  df-oprab 5503  df-mpt2 5504  df-pnf 7042  df-mnf 7043  df-xr 7044  df-ltxr 7045  df-le 7046  df-sub 7164  df-neg 7165  df-rp 8555  df-icc 8731
This theorem is referenced by:  iccf1o  8839
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