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Theorem icccvx 18448
Description: A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
icccvx  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( A [,] B
) ) )

Proof of Theorem icccvx
StepHypRef Expression
1 iccssre 10731 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
21sselda 3180 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR )
32adantrr 697 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  C  e.  RR )
41sselda 3180 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  RR )
54adantrl 696 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  D  e.  RR )
63, 5lttri4d 8960 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  <  D  \/  C  =  D  \/  D  <  C ) )
763adantr3 1116 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( C  <  D  \/  C  =  D  \/  D  <  C ) )
8 iccss2 10720 . . . . . . . 8  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( C [,] D
)  C_  ( A [,] B ) )
98adantl 452 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
1093adantr3 1116 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
1110adantr 451 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  C  <  D )  -> 
( C [,] D
)  C_  ( A [,] B ) )
123, 5jca 518 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  e.  RR  /\  D  e.  RR ) )
13123adantr3 1116 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( C  e.  RR  /\  D  e.  RR ) )
14 simpr3 963 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  T  e.  ( 0 [,] 1
) )
1513, 14jca 518 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  (
( C  e.  RR  /\  D  e.  RR )  /\  T  e.  ( 0 [,] 1 ) ) )
16 lincmb01cmp 10777 . . . . . . . . . 10  |-  ( ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( C [,] D ) )
1716ex 423 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D ) ) )
18173expa 1151 . . . . . . . 8  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D
) ) )
1918imp 418 . . . . . . 7  |-  ( ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( C [,] D ) )
2019an32s 779 . . . . . 6  |-  ( ( ( ( C  e.  RR  /\  D  e.  RR )  /\  T  e.  ( 0 [,] 1
) )  /\  C  <  D )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( C [,] D ) )
2115, 20sylan 457 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  C  <  D )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D ) )
2211, 21sseldd 3181 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  C  <  D )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( A [,] B ) )
23 oveq2 5866 . . . . . . 7  |-  ( C  =  D  ->  (
( 1  -  T
)  x.  C )  =  ( ( 1  -  T )  x.  D ) )
2423oveq1d 5873 . . . . . 6  |-  ( C  =  D  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
25 0re 8838 . . . . . . . . . . . 12  |-  0  e.  RR
26 1re 8837 . . . . . . . . . . . 12  |-  1  e.  RR
27 iccssre 10731 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
2825, 26, 27mp2an 653 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  C_  RR
2928sseli 3176 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
3029recnd 8861 . . . . . . . . 9  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  CC )
3130ad2antll 709 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  CC )
324recnd 8861 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  CC )
3332adantrr 697 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  D  e.  CC )
34 ax-1cn 8795 . . . . . . . . . . . 12  |-  1  e.  CC
35 npcan 9060 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3634, 35mpan 651 . . . . . . . . . . 11  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
3736adantr 451 . . . . . . . . . 10  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3837oveq1d 5873 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( 1  x.  D ) )
39 subcl 9051 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
4034, 39mpan 651 . . . . . . . . . . 11  |-  ( T  e.  CC  ->  (
1  -  T )  e.  CC )
4140ancri 535 . . . . . . . . . 10  |-  ( T  e.  CC  ->  (
( 1  -  T
)  e.  CC  /\  T  e.  CC )
)
42 adddir 8830 . . . . . . . . . . 11  |-  ( ( ( 1  -  T
)  e.  CC  /\  T  e.  CC  /\  D  e.  CC )  ->  (
( ( 1  -  T )  +  T
)  x.  D )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
43423expa 1151 . . . . . . . . . 10  |-  ( ( ( ( 1  -  T )  e.  CC  /\  T  e.  CC )  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D )  =  ( ( ( 1  -  T )  x.  D
)  +  ( T  x.  D ) ) )
4441, 43sylan 457 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( ( ( 1  -  T
)  x.  D )  +  ( T  x.  D ) ) )
45 mulid2 8836 . . . . . . . . . 10  |-  ( D  e.  CC  ->  (
1  x.  D )  =  D )
4645adantl 452 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( 1  x.  D
)  =  D )
4738, 44, 463eqtr3d 2323 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
4831, 33, 47syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
49483adantr1 1114 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  (
( ( 1  -  T )  x.  D
)  +  ( T  x.  D ) )  =  D )
5024, 49sylan9eqr 2337 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  C  =  D )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  =  D )
51 simplr2 998 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  C  =  D )  ->  D  e.  ( A [,] B ) )
5250, 51eqeltrd 2357 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  C  =  D )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( A [,] B ) )
53 iccss2 10720 . . . . . . . . 9  |-  ( ( D  e.  ( A [,] B )  /\  C  e.  ( A [,] B ) )  -> 
( D [,] C
)  C_  ( A [,] B ) )
5453adantl 452 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  C  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
5554ancom2s 777 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
56553adantr3 1116 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
5756adantr 451 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  D  <  C )  -> 
( D [,] C
)  C_  ( A [,] B ) )
585, 3jca 518 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D  e.  RR  /\  C  e.  RR ) )
59583adantr3 1116 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( D  e.  RR  /\  C  e.  RR ) )
6059, 14jca 518 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  (
( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) ) )
61 iirev 18427 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
6228, 61sseldi 3178 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  RR )
6362recnd 8861 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  CC )
64 recn 8827 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  CC )
65 mulcl 8821 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  -  T
)  e.  CC  /\  C  e.  CC )  ->  ( ( 1  -  T )  x.  C
)  e.  CC )
6663, 64, 65syl2anr 464 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  C )  e.  CC )
6766adantll 694 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
1  -  T )  x.  C )  e.  CC )
68 recn 8827 . . . . . . . . . . . . . . . 16  |-  ( D  e.  RR  ->  D  e.  CC )
69 mulcl 8821 . . . . . . . . . . . . . . . 16  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( T  x.  D
)  e.  CC )
7030, 68, 69syl2anr 464 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
7170adantlr 695 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
7267, 71addcomd 9014 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  =  ( ( T  x.  D )  +  ( ( 1  -  T
)  x.  C ) ) )
73723adantl3 1113 . . . . . . . . . . . 12  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  =  ( ( T  x.  D
)  +  ( ( 1  -  T )  x.  C ) ) )
74 nncan 9076 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
7534, 74mpan 651 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
1  -  ( 1  -  T ) )  =  T )
7675eqcomd 2288 . . . . . . . . . . . . . . . 16  |-  ( T  e.  CC  ->  T  =  ( 1  -  ( 1  -  T
) ) )
7776oveq1d 5873 . . . . . . . . . . . . . . 15  |-  ( T  e.  CC  ->  ( T  x.  D )  =  ( ( 1  -  ( 1  -  T ) )  x.  D ) )
7877oveq1d 5873 . . . . . . . . . . . . . 14  |-  ( T  e.  CC  ->  (
( T  x.  D
)  +  ( ( 1  -  T )  x.  C ) )  =  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) ) )
7930, 78syl 15 . . . . . . . . . . . . 13  |-  ( T  e.  ( 0 [,] 1 )  ->  (
( T  x.  D
)  +  ( ( 1  -  T )  x.  C ) )  =  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) ) )
8079adantl 452 . . . . . . . . . . . 12  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( T  x.  D )  +  ( ( 1  -  T )  x.  C
) )  =  ( ( ( 1  -  ( 1  -  T
) )  x.  D
)  +  ( ( 1  -  T )  x.  C ) ) )
8173, 80eqtrd 2315 . . . . . . . . . . 11  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  =  ( ( ( 1  -  ( 1  -  T
) )  x.  D
)  +  ( ( 1  -  T )  x.  C ) ) )
82 lincmb01cmp 10777 . . . . . . . . . . . 12  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  ( 1  -  T
)  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) )  e.  ( D [,] C ) )
8361, 82sylan2 460 . . . . . . . . . . 11  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) )  e.  ( D [,] C ) )
8481, 83eqeltrd 2357 . . . . . . . . . 10  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( D [,] C ) )
8584ex 423 . . . . . . . . 9  |-  ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C ) ) )
86853expa 1151 . . . . . . . 8  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C
) ) )
8786imp 418 . . . . . . 7  |-  ( ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( D [,] C ) )
8887an32s 779 . . . . . 6  |-  ( ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1
) )  /\  D  <  C )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( D [,] C ) )
8960, 88sylan 457 . . . . 5  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  D  <  C )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C ) )
9057, 89sseldd 3181 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  D  <  C )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( A [,] B ) )
9122, 52, 903jaodan 1248 . . 3  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  ( C  <  D  \/  C  =  D  \/  D  <  C ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( A [,] B ) )
927, 91mpdan 649 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( A [,] B ) )
9392ex 423 1  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( A [,] B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    < clt 8867    - cmin 9037   [,]cicc 10659
This theorem is referenced by:  reparphti  18495
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-rp 10355  df-icc 10663
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