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Theorem icccvx 21976
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 11712 . . . . . . 7  |-  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) )  -> 
( C [,] D
)  C_  ( A [,] B ) )
21adantl 467 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C [,] D )  C_  ( A [,] B ) )
323adantr3 1166 . . . . 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 466 . . . 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 11723 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
65sselda 3464 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  C  e.  ( A [,] B ) )  ->  C  e.  RR )
76adantrr 721 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  C  e.  RR )
85sselda 3464 . . . . . . . . 9  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  RR )
98adantrl 720 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  D  e.  RR )
107, 9jca 534 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  e.  RR  /\  D  e.  RR ) )
11103adantr3 1166 . . . . . 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 1013 . . . . . 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 534 . . . . 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 11782 . . . . . . . . 9  |-  ( ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( C [,] D ) )
1514ex 435 . . . . . . . 8  |-  ( ( C  e.  RR  /\  D  e.  RR  /\  C  <  D )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D ) ) )
16153expa 1205 . . . . . . 7  |-  ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( C [,] D
) ) )
1716imp 430 . . . . . 6  |-  ( ( ( ( C  e.  RR  /\  D  e.  RR )  /\  C  <  D )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( C [,] D ) )
1817an32s 811 . . . . 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 473 . . . 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 3465 . . 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 6313 . . . . . 6  |-  ( C  =  D  ->  (
( 1  -  T
)  x.  C )  =  ( ( 1  -  T )  x.  D ) )
2221oveq1d 6320 . . . . 5  |-  ( C  =  D  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
23 unitssre 11786 . . . . . . . . . 10  |-  ( 0 [,] 1 )  C_  RR
2423sseli 3460 . . . . . . . . 9  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  RR )
2524recnd 9676 . . . . . . . 8  |-  ( T  e.  ( 0 [,] 1 )  ->  T  e.  CC )
2625ad2antll 733 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  T  e.  CC )
278recnd 9676 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  D  e.  ( A [,] B ) )  ->  D  e.  CC )
2827adantrr 721 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  ->  D  e.  CC )
29 ax-1cn 9604 . . . . . . . . . . 11  |-  1  e.  CC
30 npcan 9891 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3129, 30mpan 674 . . . . . . . . . 10  |-  ( T  e.  CC  ->  (
( 1  -  T
)  +  T )  =  1 )
3231adantr 466 . . . . . . . . 9  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( 1  -  T )  +  T
)  =  1 )
3332oveq1d 6320 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( 1  x.  D ) )
34 subcl 9881 . . . . . . . . . . 11  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  T
)  e.  CC )
3529, 34mpan 674 . . . . . . . . . 10  |-  ( T  e.  CC  ->  (
1  -  T )  e.  CC )
3635ancri 554 . . . . . . . . 9  |-  ( T  e.  CC  ->  (
( 1  -  T
)  e.  CC  /\  T  e.  CC )
)
37 adddir 9641 . . . . . . . . . 10  |-  ( ( ( 1  -  T
)  e.  CC  /\  T  e.  CC  /\  D  e.  CC )  ->  (
( ( 1  -  T )  +  T
)  x.  D )  =  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D
) ) )
38373expa 1205 . . . . . . . . 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 473 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  +  T )  x.  D
)  =  ( ( ( 1  -  T
)  x.  D )  +  ( T  x.  D ) ) )
40 mulid2 9648 . . . . . . . . 9  |-  ( D  e.  CC  ->  (
1  x.  D )  =  D )
4140adantl 467 . . . . . . . 8  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( 1  x.  D
)  =  D )
4233, 39, 413eqtr3d 2471 . . . . . . 7  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
4326, 28, 42syl2anc 665 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  T  e.  ( 0 [,] 1
) ) )  -> 
( ( ( 1  -  T )  x.  D )  +  ( T  x.  D ) )  =  D )
44433adantr1 1164 . . . . 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 2485 . . . 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 1048 . . . 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 2507 . . 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 11712 . . . . . . . 8  |-  ( ( D  e.  ( A [,] B )  /\  C  e.  ( A [,] B ) )  -> 
( D [,] C
)  C_  ( A [,] B ) )
4948adantl 467 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( D  e.  ( A [,] B
)  /\  C  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
5049ancom2s 809 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D [,] C )  C_  ( A [,] B ) )
51503adantr3 1166 . . . . 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 466 . . . 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 534 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( D  e.  RR  /\  C  e.  RR ) )
54533adantr3 1166 . . . . . 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 534 . . . . 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 21955 . . . . . . . . . . . . . . . . 17  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
5723, 56sseldi 3462 . . . . . . . . . . . . . . . 16  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  RR )
5857recnd 9676 . . . . . . . . . . . . . . 15  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  CC )
59 recn 9636 . . . . . . . . . . . . . . 15  |-  ( C  e.  RR  ->  C  e.  CC )
60 mulcl 9630 . . . . . . . . . . . . . . 15  |-  ( ( ( 1  -  T
)  e.  CC  /\  C  e.  CC )  ->  ( ( 1  -  T )  x.  C
)  e.  CC )
6158, 59, 60syl2anr 480 . . . . . . . . . . . . . 14  |-  ( ( C  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( 1  -  T )  x.  C )  e.  CC )
6261adantll 718 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( (
1  -  T )  x.  C )  e.  CC )
63 recn 9636 . . . . . . . . . . . . . . 15  |-  ( D  e.  RR  ->  D  e.  CC )
64 mulcl 9630 . . . . . . . . . . . . . . 15  |-  ( ( T  e.  CC  /\  D  e.  CC )  ->  ( T  x.  D
)  e.  CC )
6525, 63, 64syl2anr 480 . . . . . . . . . . . . . 14  |-  ( ( D  e.  RR  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
6665adantlr 719 . . . . . . . . . . . . 13  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( T  x.  D )  e.  CC )
6762, 66addcomd 9842 . . . . . . . . . . . 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 1163 . . . . . . . . . . 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 9910 . . . . . . . . . . . . . . . . 17  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
7029, 69mpan 674 . . . . . . . . . . . . . . . 16  |-  ( T  e.  CC  ->  (
1  -  ( 1  -  T ) )  =  T )
7170eqcomd 2430 . . . . . . . . . . . . . . 15  |-  ( T  e.  CC  ->  T  =  ( 1  -  ( 1  -  T
) ) )
7271oveq1d 6320 . . . . . . . . . . . . . 14  |-  ( T  e.  CC  ->  ( T  x.  D )  =  ( ( 1  -  ( 1  -  T ) )  x.  D ) )
7372oveq1d 6320 . . . . . . . . . . . . 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 467 . . . . . . . . . . 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 2463 . . . . . . . . . 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 11782 . . . . . . . . . . 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 476 . . . . . . . . . 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 2507 . . . . . . . . 9  |-  ( ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  /\  T  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D
) )  e.  ( D [,] C ) )
8079ex 435 . . . . . . . 8  |-  ( ( D  e.  RR  /\  C  e.  RR  /\  D  <  C )  ->  ( T  e.  ( 0 [,] 1 )  -> 
( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C ) ) )
81803expa 1205 . . . . . . 7  |-  ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C
)  ->  ( T  e.  ( 0 [,] 1
)  ->  ( (
( 1  -  T
)  x.  C )  +  ( T  x.  D ) )  e.  ( D [,] C
) ) )
8281imp 430 . . . . . 6  |-  ( ( ( ( D  e.  RR  /\  C  e.  RR )  /\  D  <  C )  /\  T  e.  ( 0 [,] 1
) )  ->  (
( ( 1  -  T )  x.  C
)  +  ( T  x.  D ) )  e.  ( D [,] C ) )
8382an32s 811 . . . . 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 473 . . . 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 3465 . . 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 9783 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( C  e.  ( A [,] B
)  /\  D  e.  ( A [,] B ) ) )  ->  ( C  <  D  \/  C  =  D  \/  D  <  C ) )
87863adantr3 1166 . . 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 1331 . 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 435 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 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1872    C_ wss 3436   class class class wbr 4423  (class class class)co 6305   CCcc 9544   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549    x. cmul 9551    < clt 9682    - cmin 9867   [,]cicc 11645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-id 4768  df-po 4774  df-so 4775  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-er 7374  df-en 7581  df-dom 7582  df-sdom 7583  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-rp 11310  df-icc 11649
This theorem is referenced by:  reparphti  22026
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