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Theorem cvxcl 20111
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 453 . . . 4  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  D  C_  RR )
3 simpr1 966 . . . 4  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  X  e.  D )
42, 3sseldd 3104 . . 3  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  X  e.  RR )
5 simpr2 967 . . . 4  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  Y  e.  D )
62, 5sseldd 3104 . . 3  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  Y  e.  RR )
74, 6lttri4d 8840 . 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 2597 . . . . . 6  |-  ( ph  ->  A. x  e.  D  A. y  e.  D  ( x [,] y
)  C_  D )
109ad2antrr 709 . . . . 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 5717 . . . . . . . . 9  |-  ( x  =  X  ->  (
x [,] y )  =  ( X [,] y ) )
1211sseq1d 3126 . . . . . . . 8  |-  ( x  =  X  ->  (
( x [,] y
)  C_  D  <->  ( X [,] y )  C_  D
) )
13 oveq2 5718 . . . . . . . . 9  |-  ( y  =  Y  ->  ( X [,] y )  =  ( X [,] Y
) )
1413sseq1d 3126 . . . . . . . 8  |-  ( y  =  Y  ->  (
( X [,] y
)  C_  D  <->  ( X [,] Y )  C_  D
) )
1512, 14rcla42v 2827 . . . . . . 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 645 . . . . . 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 453 . . . . 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 16 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  ( X [,] Y )  C_  D )
19 ax-1cn 8675 . . . . . . . . 9  |-  1  e.  CC
20 0re 8718 . . . . . . . . . . . 12  |-  0  e.  RR
21 1re 8717 . . . . . . . . . . . 12  |-  1  e.  RR
22 iccssre 10609 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
2320, 21, 22mp2an 656 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  C_  RR
24 simpr3 968 . . . . . . . . . . 11  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  ( 0 [,] 1 ) )
2523, 24sseldi 3101 . . . . . . . . . 10  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  RR )
2625recnd 8741 . . . . . . . . 9  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  CC )
27 nncan 8956 . . . . . . . . 9  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
2819, 26, 27sylancr 647 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  -  (
1  -  T ) )  =  T )
2928oveq1d 5725 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( 1  -  ( 1  -  T
) )  x.  X
)  =  ( T  x.  X ) )
3029oveq1d 5725 . . . . . 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 453 . . . . 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 453 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  X  e.  RR )
336adantr 453 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  Y  e.  RR )
34 simpr 449 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  X  <  Y )
35 simplr3 1004 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  T  e.  ( 0 [,] 1
) )
36 iirev 18259 . . . . . . 7  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
3735, 36syl 17 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  < 
Y )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
38 lincmb01cmp 10655 . . . . . 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 1190 . . . . 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 2328 . . . 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 3104 . . 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 5718 . . . . . 6  |-  ( X  =  Y  ->  ( T  x.  X )  =  ( T  x.  Y ) )
4342oveq1d 5725 . . . . 5  |-  ( X  =  Y  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  =  ( ( T  x.  Y )  +  ( ( 1  -  T )  x.  Y
) ) )
44 pncan3 8939 . . . . . . . 8  |-  ( ( T  e.  CC  /\  1  e.  CC )  ->  ( T  +  ( 1  -  T ) )  =  1 )
4526, 19, 44sylancl 646 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( T  +  ( 1  -  T ) )  =  1 )
4645oveq1d 5725 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  +  ( 1  -  T
) )  x.  Y
)  =  ( 1  x.  Y ) )
47 resubcl 8991 . . . . . . . . 9  |-  ( ( 1  e.  RR  /\  T  e.  RR )  ->  ( 1  -  T
)  e.  RR )
4821, 25, 47sylancr 647 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  -  T
)  e.  RR )
4948recnd 8741 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  -  T
)  e.  CC )
506recnd 8741 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  Y  e.  CC )
5126, 49, 50adddird 8740 . . . . . 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 8733 . . . . . 6  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( 1  x.  Y
)  =  Y )
5346, 51, 523eqtr3d 2293 . . . . 5  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( T  x.  Y )  +  ( ( 1  -  T
)  x.  Y ) )  =  Y )
5443, 53sylan9eqr 2307 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  =  Y )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  =  Y )
555adantr 453 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  =  Y )  ->  Y  e.  D )
5654, 55eqeltrd 2327 . . 3  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  X  =  Y )  ->  (
( T  x.  X
)  +  ( ( 1  -  T )  x.  Y ) )  e.  D )
579ad2antrr 709 . . . . 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 5717 . . . . . . . . 9  |-  ( x  =  Y  ->  (
x [,] y )  =  ( Y [,] y ) )
5958sseq1d 3126 . . . . . . . 8  |-  ( x  =  Y  ->  (
( x [,] y
)  C_  D  <->  ( Y [,] y )  C_  D
) )
60 oveq2 5718 . . . . . . . . 9  |-  ( y  =  X  ->  ( Y [,] y )  =  ( Y [,] X
) )
6160sseq1d 3126 . . . . . . . 8  |-  ( y  =  X  ->  (
( Y [,] y
)  C_  D  <->  ( Y [,] X )  C_  D
) )
6259, 61rcla42v 2827 . . . . . . 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 645 . . . . . 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 453 . . . . 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 16 . . . 4  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  ( Y [,] X )  C_  D )
664recnd 8741 . . . . . . . 8  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  ->  X  e.  CC )
6726, 66mulcld 8735 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( T  x.  X
)  e.  CC )
6849, 50mulcld 8735 . . . . . . 7  |-  ( (
ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( 1  -  T )  x.  Y
)  e.  CC )
6967, 68addcomd 8894 . . . . . 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 453 . . . . 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 453 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  Y  e.  RR )
724adantr 453 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  X  e.  RR )
73 simpr 449 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  Y  <  X )
74 simplr3 1004 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  D  /\  Y  e.  D  /\  T  e.  ( 0 [,] 1 ) ) )  /\  Y  < 
X )  ->  T  e.  ( 0 [,] 1
) )
75 lincmb01cmp 10655 . . . . . 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 1190 . . . . 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 2327 . . . 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 3104 . . 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 1253 . 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 652 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 6    /\ wa 360    \/ w3o 938    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2509    C_ wss 3078   class class class wbr 3920  (class class class)co 5710   CCcc 8615   RRcr 8616   0cc0 8617   1c1 8618    + caddc 8620    x. cmul 8622    < clt 8747    - cmin 8917   [,]cicc 10537
This theorem is referenced by:  scvxcvx  20112  jensenlem2  20114  amgmlem  20116
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-po 4207  df-so 4208  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-iota 6143  df-riota 6190  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-rp 10234  df-icc 10541
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