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Theorem lincmb01cmp 11449
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 9407 . . . . . . 7  |-  0  e.  RR
32a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  0  e.  RR )
4 1re 9406 . . . . . . 7  |-  1  e.  RR
54a1i 11 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  RR )
62, 4elicc2i 11382 . . . . . . . 8  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
76simp1bi 1003 . . . . . . 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 11045 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
109biimp3a 1318 . . . . . . 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 11444 . . . . . 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 1219 . . . . 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 992 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
18 simpl1 991 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1917, 18resubcld 9797 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
2019recnd 9433 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  CC )
2120mul02d 9588 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  x.  ( B  -  A
) )  =  0 )
2220mulid2d 9425 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  ( B  -  A
) )  =  ( B  -  A ) )
2321, 22oveq12d 6130 . . . 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 2519 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  ( 0 [,] ( B  -  A ) ) )
258, 19remulcld 9435 . . . 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 11440 . . . 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 1219 . . 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 9433 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  CC )
3217recnd 9433 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
3331, 32mulcld 9427 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  B )  e.  CC )
3418recnd 9433 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
3531, 34mulcld 9427 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  A )  e.  CC )
3633, 35, 34subadd23d 9762 . . 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 9821 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  =  ( ( T  x.  B
)  -  ( T  x.  A ) ) )
3837oveq1d 6127 . . 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 9694 . . . . . . . 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 9435 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  RR )
4241recnd 9433 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  CC )
4342, 33addcomd 9592 . . . 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 ax-1cn 9361 . . . . . . . 8  |-  1  e.  CC
4544a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
4645, 31, 34subdird 9822 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( ( 1  x.  A
)  -  ( T  x.  A ) ) )
4734mulid2d 9425 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
4847oveq1d 6127 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  x.  A )  -  ( T  x.  A
) )  =  ( A  -  ( T  x.  A ) ) )
4946, 48eqtrd 2475 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( A  -  ( T  x.  A ) ) )
5049oveq2d 6128 . . . 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 ) ) ) )
5143, 50eqtrd 2475 . . 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 ) ) ) )
5236, 38, 513eqtr4d 2485 . 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 ) ) )
5334addid2d 9591 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  +  A )  =  A )
5432, 34npcand 9744 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( B  -  A )  +  A )  =  B )
5553, 54oveq12d 6130 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 0  +  A ) [,] ( ( B  -  A )  +  A
) )  =  ( A [,] B ) )
5630, 52, 553eltr3d 2523 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 965    e. wcel 1756   class class class wbr 4313  (class class class)co 6112   CCcc 9301   RRcr 9302   0cc0 9303   1c1 9304    + caddc 9306    x. cmul 9308    < clt 9439    <_ cle 9440    - cmin 9616   RR+crp 11012   [,]cicc 11324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-er 7122  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-rp 11013  df-icc 11328
This theorem is referenced by:  iccf1o  11450  icccvx  20544  efcvx  21936  logccv  22130  cvxcl  22400
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