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Theorem icccvx 18464
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 10747 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
21sselda 3193 . . . . . 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 3193 . . . . . 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 8976 . . . 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 10736 . . . . . . . 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 10793 . . . . . . . . . 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 3194 . . . 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 5882 . . . . . . 7  |-  ( C  =  D  ->  (
( 1  -  T
)  x.  C )  =  ( ( 1  -  T )  x.  D ) )
2423oveq1d 5889 . . . . . 6  |-  ( C  =  D  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
25 0re 8854 . . . . . . . . . . . 12  |-  0  e.  RR
26 1re 8853 . . . . . . . . . . . 12  |-  1  e.  RR
27 iccssre 10747 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR  /\  1  e.  RR )  ->  ( 0 [,] 1
)  C_  RR )
2825, 26, 27mp2an 653 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  C_  RR
2928sseli 3189 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
3029recnd 8877 . . . . . . . . 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 8877 . . . . . . . . 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 8811 . . . . . . . . . . . 12  |-  1  e.  CC
35 npcan 9076 . . . . . . . . . . . 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 5889 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( 1  x.  D ) )
39 subcl 9067 . . . . . . . . . . . 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 8846 . . . . . . . . . . 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 8852 . . . . . . . . . 10  |-  ( D  e.  CC  ->  (
1  x.  D )  =  D )
4645adantl 452 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( 1  x.  D
)  =  D )
4738, 44, 463eqtr3d 2336 . . . . . . . 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 2350 . . . . 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 2370 . . . 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 10736 . . . . . . . . 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 18443 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
6228, 61sseldi 3191 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  RR )
6362recnd 8877 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  CC )
64 recn 8843 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  CC )
65 mulcl 8837 . . . . . . . . . . . . . . . 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 8843 . . . . . . . . . . . . . . . 16  |-  ( D  e.  RR  ->  D  e.  CC )
69 mulcl 8837 . . . . . . . . . . . . . . . 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 9030 . . . . . . . . . . . . 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 9092 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
7534, 74mpan 651 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
1  -  ( 1  -  T ) )  =  T )
7675eqcomd 2301 . . . . . . . . . . . . . . . 16  |-  ( T  e.  CC  ->  T  =  ( 1  -  ( 1  -  T
) ) )
7776oveq1d 5889 . . . . . . . . . . . . . . 15  |-  ( T  e.  CC  ->  ( T  x.  D )  =  ( ( 1  -  ( 1  -  T ) )  x.  D ) )
7877oveq1d 5889 . . . . . . . . . . . . . 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 2328 . . . . . . . . . . 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 10793 . . . . . . . . . . . 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 2370 . . . . . . . . . 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 3194 . . . 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 1632    e. wcel 1696    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:  reparphti  18511
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-iun 3923  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-1st 6138  df-2nd 6139  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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