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Theorem icccvx 21740
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 iccss2 11647 . . . . . . 7  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( C [,] D
)  C_  ( A [,] B ) )
21adantl 464 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
323adantr3 1158 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
43adantr 463 . . . 4  |-  ( ( ( ( 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 ) )
5 iccssre 11658 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
65sselda 3441 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR )
76adantrr 715 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  C  e.  RR )
85sselda 3441 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  RR )
98adantrl 714 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  D  e.  RR )
107, 9jca 530 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  e.  RR  /\  D  e.  RR ) )
11103adantr3 1158 . . . . . 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 ) )
12 simpr3 1005 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  T  e.  ( 0 [,] 1
) )
1311, 12jca 530 . . . . 5  |-  ( ( ( 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 ) ) )
14 lincmb01cmp 11715 . . . . . . . . 9  |-  ( ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( C [,] D ) )
1514ex 432 . . . . . . . 8  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D ) ) )
16153expa 1197 . . . . . . 7  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D
) ) )
1716imp 427 . . . . . 6  |-  ( ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( C [,] D ) )
1817an32s 805 . . . . 5  |-  ( ( ( ( C  e.  RR  /\  D  e.  RR )  /\  T  e.  ( 0 [,] 1
) )  /\  C  <  D )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( C [,] D ) )
1913, 18sylan 469 . . . 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.  ( C [,] D ) )
204, 19sseldd 3442 . . 3  |-  ( ( ( ( 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 ) )
21 oveq2 6285 . . . . . 6  |-  ( C  =  D  ->  (
( 1  -  T
)  x.  C )  =  ( ( 1  -  T )  x.  D ) )
2221oveq1d 6292 . . . . 5  |-  ( C  =  D  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
23 unitssre 11719 . . . . . . . . . 10  |-  ( 0 [,] 1 )  C_  RR
2423sseli 3437 . . . . . . . . 9  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
2524recnd 9651 . . . . . . . 8  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  CC )
2625ad2antll 727 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  CC )
278recnd 9651 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  CC )
2827adantrr 715 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  D  e.  CC )
29 ax-1cn 9579 . . . . . . . . . . 11  |-  1  e.  CC
30 npcan 9864 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3129, 30mpan 668 . . . . . . . . . 10  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
3231adantr 463 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3332oveq1d 6292 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( 1  x.  D ) )
34 subcl 9854 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
3529, 34mpan 668 . . . . . . . . . 10  |-  ( T  e.  CC  ->  (
1  -  T )  e.  CC )
3635ancri 550 . . . . . . . . 9  |-  ( T  e.  CC  ->  (
( 1  -  T
)  e.  CC  /\  T  e.  CC )
)
37 adddir 9616 . . . . . . . . . 10  |-  ( ( ( 1  -  T
)  e.  CC  /\  T  e.  CC  /\  D  e.  CC )  ->  (
( ( 1  -  T )  +  T
)  x.  D )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
38373expa 1197 . . . . . . . . 9  |-  ( ( ( ( 1  -  T )  e.  CC  /\  T  e.  CC )  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D )  =  ( ( ( 1  -  T )  x.  D
)  +  ( T  x.  D ) ) )
3936, 38sylan 469 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( ( ( 1  -  T
)  x.  D )  +  ( T  x.  D ) ) )
40 mulid2 9623 . . . . . . . . 9  |-  ( D  e.  CC  ->  (
1  x.  D )  =  D )
4140adantl 464 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( 1  x.  D
)  =  D )
4233, 39, 413eqtr3d 2451 . . . . . . 7  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
4326, 28, 42syl2anc 659 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
44433adantr1 1156 . . . . 5  |-  ( ( ( 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 )
4522, 44sylan9eqr 2465 . . . 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 ) )  =  D )
46 simplr2 1040 . . . 4  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  /\  C  =  D )  ->  D  e.  ( A [,] B ) )
4745, 46eqeltrd 2490 . . 3  |-  ( ( ( ( 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 ) )
48 iccss2 11647 . . . . . . . 8  |-  ( ( D  e.  ( A [,] B )  /\  C  e.  ( A [,] B ) )  -> 
( D [,] C
)  C_  ( A [,] B ) )
4948adantl 464 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  C  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
5049ancom2s 803 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
51503adantr3 1158 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B )  /\  T  e.  ( 0 [,] 1 ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
5251adantr 463 . . . 4  |-  ( ( ( ( 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 ) )
539, 7jca 530 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D  e.  RR  /\  C  e.  RR ) )
54533adantr3 1158 . . . . . 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 ) )
5554, 12jca 530 . . . . 5  |-  ( ( ( 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 ) ) )
56 iirev 21719 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
5723, 56sseldi 3439 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  RR )
5857recnd 9651 . . . . . . . . . . . . . . 15  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  CC )
59 recn 9611 . . . . . . . . . . . . . . 15  |-  ( C  e.  RR  ->  C  e.  CC )
60 mulcl 9605 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  -  T
)  e.  CC  /\  C  e.  CC )  ->  ( ( 1  -  T )  x.  C
)  e.  CC )
6158, 59, 60syl2anr 476 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  C )  e.  CC )
6261adantll 712 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
1  -  T )  x.  C )  e.  CC )
63 recn 9611 . . . . . . . . . . . . . . 15  |-  ( D  e.  RR  ->  D  e.  CC )
64 mulcl 9605 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( T  x.  D
)  e.  CC )
6525, 63, 64syl2anr 476 . . . . . . . . . . . . . 14  |-  ( ( D  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
6665adantlr 713 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
6762, 66addcomd 9815 . . . . . . . . . . . 12  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  =  ( ( T  x.  D )  +  ( ( 1  -  T
)  x.  C ) ) )
68673adantl3 1155 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) )
69 nncan 9883 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
7029, 69mpan 668 . . . . . . . . . . . . . . . 16  |-  ( T  e.  CC  ->  (
1  -  ( 1  -  T ) )  =  T )
7170eqcomd 2410 . . . . . . . . . . . . . . 15  |-  ( T  e.  CC  ->  T  =  ( 1  -  ( 1  -  T
) ) )
7271oveq1d 6292 . . . . . . . . . . . . . 14  |-  ( T  e.  CC  ->  ( T  x.  D )  =  ( ( 1  -  ( 1  -  T ) )  x.  D ) )
7372oveq1d 6292 . . . . . . . . . . . . 13  |-  ( T  e.  CC  ->  (
( T  x.  D
)  +  ( ( 1  -  T )  x.  C ) )  =  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) ) )
7425, 73syl 17 . . . . . . . . . . . 12  |-  ( T  e.  ( 0 [,] 1 )  ->  (
( T  x.  D
)  +  ( ( 1  -  T )  x.  C ) )  =  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) ) )
7574adantl 464 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) )
7668, 75eqtrd 2443 . . . . . . . . . 10  |-  ( ( ( 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 ) ) )
77 lincmb01cmp 11715 . . . . . . . . . . 11  |-  ( ( ( 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 ) )
7856, 77sylan2 472 . . . . . . . . . 10  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  D )  +  ( ( 1  -  T )  x.  C
) )  e.  ( D [,] C ) )
7976, 78eqeltrd 2490 . . . . . . . . 9  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( D [,] C ) )
8079ex 432 . . . . . . . 8  |-  ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C ) ) )
81803expa 1197 . . . . . . 7  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C
) ) )
8281imp 427 . . . . . 6  |-  ( ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( D [,] C ) )
8382an32s 805 . . . . 5  |-  ( ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1
) )  /\  D  <  C )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( D [,] C ) )
8455, 83sylan 469 . . . 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.  ( D [,] C ) )
8552, 84sseldd 3442 . . 3  |-  ( ( ( ( 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 ) )
867, 9lttri4d 9757 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  <  D  \/  C  =  D  \/  D  <  C ) )
87863adantr3 1158 . . 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 ) )
8820, 47, 85, 87mpjao3dan 1297 . 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 ) )
8988ex 432 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 367    \/ w3o 973    /\ w3a 974    = wceq 1405    e. wcel 1842    C_ wss 3413   class class class wbr 4394  (class class class)co 6277   CCcc 9519   RRcr 9520   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    < clt 9657    - cmin 9840   [,]cicc 11584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-rp 11265  df-icc 11588
This theorem is referenced by:  reparphti  21787
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