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Theorem lincmb01cmp 11674
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 461 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
2 0re 9599 . . . . . . 7  |-  0  e.  RR
32a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  0  e.  RR )
4 1re 9598 . . . . . . 7  |-  1  e.  RR
54a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  RR )
62, 4elicc2i 11601 . . . . . . . 8  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
76simp1bi 1012 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
87adantl 466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  RR )
9 difrp 11264 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
109biimp3a 1329 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  ->  ( B  -  A )  e.  RR+ )
1110adantr 465 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR+ )
12 eqid 2443 . . . . . . 7  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
13 eqid 2443 . . . . . . 7  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
1412, 13iccdil 11669 . . . . . 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 1230 . . . . 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 210 . . . 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 1001 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
18 simpl1 1000 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1917, 18resubcld 9994 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
2019recnd 9625 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  CC )
2120mul02d 9781 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  x.  ( B  -  A
) )  =  0 )
2220mulid2d 9617 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  ( B  -  A
) )  =  ( B  -  A ) )
2321, 22oveq12d 6299 . . . 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 2533 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  ( 0 [,] ( B  -  A ) ) )
258, 19remulcld 9627 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  RR )
26 eqid 2443 . . . . 5  |-  ( 0  +  A )  =  ( 0  +  A
)
27 eqid 2443 . . . . 5  |-  ( ( B  -  A )  +  A )  =  ( ( B  -  A )  +  A
)
2826, 27iccshftr 11665 . . . 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 1230 . . 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 210 . 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 9625 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  CC )
3217recnd 9625 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
3331, 32mulcld 9619 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  B )  e.  CC )
3418recnd 9625 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
3531, 34mulcld 9619 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  A )  e.  CC )
3633, 35, 34subadd23d 9958 . . 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 10019 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  =  ( ( T  x.  B
)  -  ( T  x.  A ) ) )
3837oveq1d 6296 . . 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 9888 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
404, 8, 39sylancr 663 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  -  T )  e.  RR )
4140, 18remulcld 9627 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  RR )
4241recnd 9625 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  CC )
4342, 33addcomd 9785 . . . 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 9615 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
4544, 31, 34subdird 10020 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( ( 1  x.  A
)  -  ( T  x.  A ) ) )
4634mulid2d 9617 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
4746oveq1d 6296 . . . . . 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 2484 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( A  -  ( T  x.  A ) ) )
4948oveq2d 6297 . . . 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 2484 . . 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 2494 . 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 9784 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  +  A )  =  A )
5332, 34npcand 9940 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( B  -  A )  +  A )  =  B )
5452, 53oveq12d 6299 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 0  +  A ) [,] ( ( B  -  A )  +  A
) )  =  ( A [,] B ) )
5530, 51, 543eltr3d 2545 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    e. wcel 1804   class class class wbr 4437  (class class class)co 6281   RRcr 9494   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500    < clt 9631    <_ cle 9632    - cmin 9810   RR+crp 11231   [,]cicc 11543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-rp 11232  df-icc 11547
This theorem is referenced by:  iccf1o  11675  icccvx  21428  efcvx  22822  logccv  23022  cvxcl  23292
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