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Theorem lincmb01cmp 9960
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 7920 . . . . . . 7  |-  0  e.  RR
32a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  0  e.  RR )
4 1re 7919 . . . . . . 7  |-  1  e.  RR
54a1i 9 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  RR )
62, 4elicc2i 9896 . . . . . . . 8  |-  ( T  e.  ( 0 [,] 1 )  <->  ( T  e.  RR  /\  0  <_  T  /\  T  <_  1
) )
76simp1bi 1007 . . . . . . 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 9649 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  B  <->  ( B  -  A )  e.  RR+ ) )
109biimp3a 1340 . . . . . . 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 2170 . . . . . . 7  |-  ( 0  x.  ( B  -  A ) )  =  ( 0  x.  ( B  -  A )
)
13 eqid 2170 . . . . . . 7  |-  ( 1  x.  ( B  -  A ) )  =  ( 1  x.  ( B  -  A )
)
1412, 13iccdil 9955 . . . . . 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 1234 . . . . 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 996 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  RR )
18 simpl1 995 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  RR )
1917, 18resubcld 8300 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  RR )
2019recnd 7948 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( B  -  A )  e.  CC )
2120mul02d 8311 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  x.  ( B  -  A
) )  =  0 )
2220mulid2d 7938 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  ( B  -  A
) )  =  ( B  -  A ) )
2321, 22oveq12d 5871 . . . 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 2249 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  ( 0 [,] ( B  -  A ) ) )
258, 19remulcld 7950 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  e.  RR )
26 eqid 2170 . . . . 5  |-  ( 0  +  A )  =  ( 0  +  A
)
27 eqid 2170 . . . . 5  |-  ( ( B  -  A )  +  A )  =  ( ( B  -  A )  +  A
)
2826, 27iccshftr 9951 . . . 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 1234 . . 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 7948 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  T  e.  CC )
3217recnd 7948 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  B  e.  CC )
3331, 32mulcld 7940 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  B )  e.  CC )
3418recnd 7948 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  A  e.  CC )
3531, 34mulcld 7940 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  A )  e.  CC )
3633, 35, 34subadd23d 8252 . . 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 8333 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  ( B  -  A
) )  =  ( ( T  x.  B
)  -  ( T  x.  A ) ) )
3837oveq1d 5868 . . 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 8183 . . . . . . . 8  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
404, 8, 39sylancr 412 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  -  T )  e.  RR )
4140, 18remulcld 7950 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  RR )
4241recnd 7948 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  e.  CC )
4342, 33addcomd 8070 . . . 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 7936 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  1  e.  CC )
4544, 31, 34subdird 8334 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( ( 1  x.  A
)  -  ( T  x.  A ) ) )
4634mulid2d 7938 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 1  x.  A )  =  A )
4746oveq1d 5868 . . . . . 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 2203 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  A )  =  ( A  -  ( T  x.  A ) ) )
4948oveq2d 5869 . . . 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 2203 . . 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 2213 . 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 8069 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( 0  +  A )  =  A )
5332, 34npcand 8234 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( B  -  A )  +  A )  =  B )
5452, 53oveq12d 5871 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 0  +  A ) [,] ( ( B  -  A )  +  A
) )  =  ( A [,] B ) )
5530, 51, 543eltr3d 2253 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 973    e. wcel 2141   class class class wbr 3989  (class class class)co 5853   RRcr 7773   0cc0 7774   1c1 7775    + caddc 7777    x. cmul 7779    < clt 7954    <_ cle 7955    - cmin 8090   RR+crp 9610   [,]cicc 9848
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890  ax-pre-mulgt0 7891
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-rp 9611  df-icc 9852
This theorem is referenced by:  iccf1o  9961
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