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Theorem icccvx 21182
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 11602 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
21sselda 3504 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR )
32adantrr 716 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  C  e.  RR )
41sselda 3504 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  RR )
54adantrl 715 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  D  e.  RR )
63, 5lttri4d 9721 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  <  D  \/  C  =  D  \/  D  <  C ) )
763adantr3 1157 . . 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 11591 . . . . . . . 8  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( C [,] D
)  C_  ( A [,] B ) )
98adantl 466 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
1093adantr3 1157 . . . . . 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 465 . . . . 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 532 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  e.  RR  /\  D  e.  RR ) )
13123adantr3 1157 . . . . . . 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 1004 . . . . . . 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 532 . . . . . 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 11659 . . . . . . . . . 10  |-  ( ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( C [,] D ) )
1716ex 434 . . . . . . . . 9  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D ) ) )
18173expa 1196 . . . . . . . 8  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D
) ) )
1918imp 429 . . . . . . 7  |-  ( ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( C [,] D ) )
2019an32s 802 . . . . . 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 471 . . . . 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 3505 . . . 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 6290 . . . . . . 7  |-  ( C  =  D  ->  (
( 1  -  T
)  x.  C )  =  ( ( 1  -  T )  x.  D ) )
2423oveq1d 6297 . . . . . 6  |-  ( C  =  D  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
25 unitssre 11663 . . . . . . . . . . 11  |-  ( 0 [,] 1 )  C_  RR
2625sseli 3500 . . . . . . . . . 10  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
2726recnd 9618 . . . . . . . . 9  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  CC )
2827ad2antll 728 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  CC )
294recnd 9618 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  CC )
3029adantrr 716 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  D  e.  CC )
31 ax-1cn 9546 . . . . . . . . . . . 12  |-  1  e.  CC
32 npcan 9825 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3331, 32mpan 670 . . . . . . . . . . 11  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
3433adantr 465 . . . . . . . . . 10  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3534oveq1d 6297 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( 1  x.  D ) )
36 subcl 9815 . . . . . . . . . . . 12  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
3731, 36mpan 670 . . . . . . . . . . 11  |-  ( T  e.  CC  ->  (
1  -  T )  e.  CC )
3837ancri 552 . . . . . . . . . 10  |-  ( T  e.  CC  ->  (
( 1  -  T
)  e.  CC  /\  T  e.  CC )
)
39 adddir 9583 . . . . . . . . . . 11  |-  ( ( ( 1  -  T
)  e.  CC  /\  T  e.  CC  /\  D  e.  CC )  ->  (
( ( 1  -  T )  +  T
)  x.  D )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
40393expa 1196 . . . . . . . . . 10  |-  ( ( ( ( 1  -  T )  e.  CC  /\  T  e.  CC )  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D )  =  ( ( ( 1  -  T )  x.  D
)  +  ( T  x.  D ) ) )
4138, 40sylan 471 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( ( ( 1  -  T
)  x.  D )  +  ( T  x.  D ) ) )
42 mulid2 9590 . . . . . . . . . 10  |-  ( D  e.  CC  ->  (
1  x.  D )  =  D )
4342adantl 466 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( 1  x.  D
)  =  D )
4435, 41, 433eqtr3d 2516 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
4528, 30, 44syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
46453adantr1 1155 . . . . . 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 )
4724, 46sylan9eqr 2530 . . . . 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 )
48 simplr2 1039 . . . . 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 ) )
4947, 48eqeltrd 2555 . . . 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 ) )
50 iccss2 11591 . . . . . . . . 9  |-  ( ( D  e.  ( A [,] B )  /\  C  e.  ( A [,] B ) )  -> 
( D [,] C
)  C_  ( A [,] B ) )
5150adantl 466 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  C  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
5251ancom2s 800 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
53523adantr3 1157 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
5453adantr 465 . . . . 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 ) )
555, 3jca 532 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D  e.  RR  /\  C  e.  RR ) )
56553adantr3 1157 . . . . . . 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 ) )
5756, 14jca 532 . . . . . 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 ) ) )
58 iirev 21161 . . . . . . . . . . . . . . . . . 18  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
5925, 58sseldi 3502 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  RR )
6059recnd 9618 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  CC )
61 recn 9578 . . . . . . . . . . . . . . . 16  |-  ( C  e.  RR  ->  C  e.  CC )
62 mulcl 9572 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1  -  T
)  e.  CC  /\  C  e.  CC )  ->  ( ( 1  -  T )  x.  C
)  e.  CC )
6360, 61, 62syl2anr 478 . . . . . . . . . . . . . . 15  |-  ( ( C  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  C )  e.  CC )
6463adantll 713 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
1  -  T )  x.  C )  e.  CC )
65 recn 9578 . . . . . . . . . . . . . . . 16  |-  ( D  e.  RR  ->  D  e.  CC )
66 mulcl 9572 . . . . . . . . . . . . . . . 16  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( T  x.  D
)  e.  CC )
6727, 65, 66syl2anr 478 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
6867adantlr 714 . . . . . . . . . . . . . 14  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
6964, 68addcomd 9777 . . . . . . . . . . . . 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 ) ) )
70693adantl3 1154 . . . . . . . . . . . 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 ) ) )
71 nncan 9844 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
7231, 71mpan 670 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  CC  ->  (
1  -  ( 1  -  T ) )  =  T )
7372eqcomd 2475 . . . . . . . . . . . . . . . 16  |-  ( T  e.  CC  ->  T  =  ( 1  -  ( 1  -  T
) ) )
7473oveq1d 6297 . . . . . . . . . . . . . . 15  |-  ( T  e.  CC  ->  ( T  x.  D )  =  ( ( 1  -  ( 1  -  T ) )  x.  D ) )
7574oveq1d 6297 . . . . . . . . . . . . . 14  |-  ( T  e.  CC  ->  (
( T  x.  D
)  +  ( ( 1  -  T )  x.  C ) )  =  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) ) )
7627, 75syl 16 . . . . . . . . . . . . 13  |-  ( T  e.  ( 0 [,] 1 )  ->  (
( T  x.  D
)  +  ( ( 1  -  T )  x.  C ) )  =  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) ) )
7776adantl 466 . . . . . . . . . . . 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 ) ) )
7870, 77eqtrd 2508 . . . . . . . . . . 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 ) ) )
79 lincmb01cmp 11659 . . . . . . . . . . . 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 ) )
8058, 79sylan2 474 . . . . . . . . . . 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 ) )
8178, 80eqeltrd 2555 . . . . . . . . . 10  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( D [,] C ) )
8281ex 434 . . . . . . . . 9  |-  ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C ) ) )
83823expa 1196 . . . . . . . 8  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C
) ) )
8483imp 429 . . . . . . 7  |-  ( ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( D [,] C ) )
8584an32s 802 . . . . . 6  |-  ( ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1
) )  /\  D  <  C )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( D [,] C ) )
8657, 85sylan 471 . . . . 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 ) )
8754, 86sseldd 3505 . . . 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 ) )
8822, 49, 873jaodan 1294 . . 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 ) )
897, 88mpdan 668 . 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 ) )
9089ex 434 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 972    /\ w3a 973    = wceq 1379    e. wcel 1767    C_ wss 3476   class class class wbr 4447  (class class class)co 6282   CCcc 9486   RRcr 9487   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    < clt 9624    - cmin 9801   [,]cicc 11528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-rp 11217  df-icc 11532
This theorem is referenced by:  reparphti  21229
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