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Theorem cvxcl 20295
Description: Closure of a 0-1 linear combination in a convex set. (Contributed by Mario Carneiro, 21-Jun-2015.)
Hypotheses
Ref Expression
cvxcl.1  |-  ( ph  ->  D  C_  RR )
cvxcl.2  |-  ( (
ph  /\  ( x  e.  D  /\  y  e.  D ) )  -> 
( x [,] y
)  C_  D )
Assertion
Ref Expression
cvxcl  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  x.  X )  +  ( ( 1  -  T
)  x.  Y ) )  e.  D )
Distinct variable groups:    x, y, D    ph, x, y    x, X, y    x, Y, y
Allowed substitution hints:    T( x, y)

Proof of Theorem cvxcl
StepHypRef Expression
1 cvxcl.1 . . . . 5  |-  ( ph  ->  D  C_  RR )
21adantr 451 . . . 4  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  D  C_  RR )
3 simpr1 961 . . . 4  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  X  e.  D )
42, 3sseldd 3194 . . 3  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  X  e.  RR )
5 simpr2 962 . . . 4  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  Y  e.  D )
62, 5sseldd 3194 . . 3  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  Y  e.  RR )
74, 6lttri4d 8976 . 2  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( X  <  Y  \/  X  =  Y  \/  Y  <  X ) )
8 cvxcl.2 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  D  /\  y  e.  D ) )  -> 
( x [,] y
)  C_  D )
98ralrimivva 2648 . . . . . 6  |-  ( ph  ->  A. x  e.  D  A. y  e.  D  ( x [,] y
)  C_  D )
109ad2antrr 706 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  A. x  e.  D  A. y  e.  D  ( x [,] y )  C_  D
)
11 oveq1 5881 . . . . . . . . 9  |-  ( x  =  X  ->  (
x [,] y )  =  ( X [,] y ) )
1211sseq1d 3218 . . . . . . . 8  |-  ( x  =  X  ->  (
( x [,] y
)  C_  D  <->  ( X [,] y )  C_  D
) )
13 oveq2 5882 . . . . . . . . 9  |-  ( y  =  Y  ->  ( X [,] y )  =  ( X [,] Y
) )
1413sseq1d 3218 . . . . . . . 8  |-  ( y  =  Y  ->  (
( X [,] y
)  C_  D  <->  ( X [,] Y )  C_  D
) )
1512, 14rspc2v 2903 . . . . . . 7  |-  ( ( X  e.  D  /\  Y  e.  D )  ->  ( A. x  e.  D  A. y  e.  D  ( x [,] y )  C_  D  ->  ( X [,] Y
)  C_  D )
)
163, 5, 15syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( A. x  e.  D  A. y  e.  D  ( x [,] y )  C_  D  ->  ( X [,] Y
)  C_  D )
)
1716adantr 451 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  ( A. x  e.  D  A. y  e.  D  ( x [,] y
)  C_  D  ->  ( X [,] Y ) 
C_  D ) )
1810, 17mpd 14 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  ( X [,] Y )  C_  D )
19 ax-1cn 8811 . . . . . . . . 9  |-  1  e.  CC
20 0re 8854 . . . . . . . . . . . 12  |-  0  e.  RR
21 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
22 iccssre 10747 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
2320, 21, 22mp2an 653 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  C_  RR
24 simpr3 963 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  ( 0 [,] 1 ) )
2523, 24sseldi 3191 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  RR )
2625recnd 8877 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  CC )
27 nncan 9092 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
2819, 26, 27sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  -  (
1  -  T ) )  =  T )
2928oveq1d 5889 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( 1  -  ( 1  -  T
) )  x.  X
)  =  ( T  x.  X ) )
3029oveq1d 5889 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  ( 1  -  T ) )  x.  X )  +  ( ( 1  -  T
)  x.  Y ) )  =  ( ( T  x.  X )  +  ( ( 1  -  T )  x.  Y ) ) )
3130adantr 451 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  (
( ( 1  -  ( 1  -  T
) )  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  =  ( ( T  x.  X )  +  ( ( 1  -  T )  x.  Y
) ) )
324adantr 451 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  X  e.  RR )
336adantr 451 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  Y  e.  RR )
34 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  X  <  Y )
35 simplr3 999 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  T  e.  ( 0 [,] 1
) )
36 iirev 18443 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
3735, 36syl 15 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
38 lincmb01cmp 10793 . . . . . 6  |-  ( ( ( X  e.  RR  /\  Y  e.  RR  /\  X  <  Y )  /\  ( 1  -  T
)  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  X )  +  ( ( 1  -  T )  x.  Y
) )  e.  ( X [,] Y ) )
3932, 33, 34, 37, 38syl31anc 1185 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  (
( ( 1  -  ( 1  -  T
) )  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  e.  ( X [,] Y ) )
4031, 39eqeltrrd 2371 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  e.  ( X [,] Y ) )
4118, 40sseldd 3194 . . 3  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  e.  D )
42 oveq2 5882 . . . . . 6  |-  ( X  =  Y  ->  ( T  x.  X )  =  ( T  x.  Y ) )
4342oveq1d 5889 . . . . 5  |-  ( X  =  Y  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  =  ( ( T  x.  Y )  +  ( ( 1  -  T )  x.  Y
) ) )
44 pncan3 9075 . . . . . . . 8  |-  ( ( T  e.  CC  /\  1  e.  CC )  ->  ( T  +  ( 1  -  T ) )  =  1 )
4526, 19, 44sylancl 643 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( T  +  ( 1  -  T ) )  =  1 )
4645oveq1d 5889 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  +  ( 1  -  T
) )  x.  Y
)  =  ( 1  x.  Y ) )
47 resubcl 9127 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
4821, 25, 47sylancr 644 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  -  T
)  e.  RR )
4948recnd 8877 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  -  T
)  e.  CC )
506recnd 8877 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  Y  e.  CC )
5126, 49, 50adddird 8876 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  +  ( 1  -  T
) )  x.  Y
)  =  ( ( T  x.  Y )  +  ( ( 1  -  T )  x.  Y ) ) )
5250mulid2d 8869 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  x.  Y
)  =  Y )
5346, 51, 523eqtr3d 2336 . . . . 5  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  x.  Y )  +  ( ( 1  -  T
)  x.  Y ) )  =  Y )
5443, 53sylan9eqr 2350 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  =  Y )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  =  Y )
555adantr 451 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  =  Y )  ->  Y  e.  D )
5654, 55eqeltrd 2370 . . 3  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  =  Y )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  e.  D )
579ad2antrr 706 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  A. x  e.  D  A. y  e.  D  ( x [,] y )  C_  D
)
58 oveq1 5881 . . . . . . . . 9  |-  ( x  =  Y  ->  (
x [,] y )  =  ( Y [,] y ) )
5958sseq1d 3218 . . . . . . . 8  |-  ( x  =  Y  ->  (
( x [,] y
)  C_  D  <->  ( Y [,] y )  C_  D
) )
60 oveq2 5882 . . . . . . . . 9  |-  ( y  =  X  ->  ( Y [,] y )  =  ( Y [,] X
) )
6160sseq1d 3218 . . . . . . . 8  |-  ( y  =  X  ->  (
( Y [,] y
)  C_  D  <->  ( Y [,] X )  C_  D
) )
6259, 61rspc2v 2903 . . . . . . 7  |-  ( ( Y  e.  D  /\  X  e.  D )  ->  ( A. x  e.  D  A. y  e.  D  ( x [,] y )  C_  D  ->  ( Y [,] X
)  C_  D )
)
635, 3, 62syl2anc 642 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( A. x  e.  D  A. y  e.  D  ( x [,] y )  C_  D  ->  ( Y [,] X
)  C_  D )
)
6463adantr 451 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  ( A. x  e.  D  A. y  e.  D  ( x [,] y
)  C_  D  ->  ( Y [,] X ) 
C_  D ) )
6557, 64mpd 14 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  ( Y [,] X )  C_  D )
664recnd 8877 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  X  e.  CC )
6726, 66mulcld 8871 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( T  x.  X
)  e.  CC )
6849, 50mulcld 8871 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( 1  -  T )  x.  Y
)  e.  CC )
6967, 68addcomd 9030 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  x.  X )  +  ( ( 1  -  T
)  x.  Y ) )  =  ( ( ( 1  -  T
)  x.  Y )  +  ( T  x.  X ) ) )
7069adantr 451 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  =  ( ( ( 1  -  T )  x.  Y )  +  ( T  x.  X
) ) )
716adantr 451 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  Y  e.  RR )
724adantr 451 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  X  e.  RR )
73 simpr 447 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  Y  <  X )
74 simplr3 999 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  T  e.  ( 0 [,] 1
) )
75 lincmb01cmp 10793 . . . . . 6  |-  ( ( ( Y  e.  RR  /\  X  e.  RR  /\  Y  <  X )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  Y )  +  ( T  x.  X
) )  e.  ( Y [,] X ) )
7671, 72, 73, 74, 75syl31anc 1185 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  (
( ( 1  -  T )  x.  Y
)  +  ( T  x.  X ) )  e.  ( Y [,] X ) )
7770, 76eqeltrd 2370 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  e.  ( Y [,] X ) )
7865, 77sseldd 3194 . . 3  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  e.  D )
7941, 56, 783jaodan 1248 . 2  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  ( X  <  Y  \/  X  =  Y  \/  Y  <  X ) )  -> 
( ( T  x.  X )  +  ( ( 1  -  T
)  x.  Y ) )  e.  D )
807, 79mpdan 649 1  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  x.  X )  +  ( ( 1  -  T
)  x.  Y ) )  e.  D )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    x. cmul 8758    < clt 8883    - cmin 9053   [,]cicc 10675
This theorem is referenced by:  scvxcvx  20296  jensenlem2  20298  amgmlem  20300
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-rp 10371  df-icc 10679
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