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Theorem efcvx 22969
Description: The exponential function on the reals is a strictly convex function. (Contributed by Mario Carneiro, 20-Jun-2015.)
Assertion
Ref Expression
efcvx  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  ( ( T  x.  ( exp `  A ) )  +  ( ( 1  -  T )  x.  ( exp `  B ) ) ) )

Proof of Theorem efcvx
StepHypRef Expression
1 simpl1 999 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR )
2 simpl2 1000 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR )
3 simpl3 1001 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <  B
)
4 reeff1o 22967 . . . . . . 7  |-  ( exp  |`  RR ) : RR -1-1-onto-> RR+
5 f1of 5822 . . . . . . 7  |-  ( ( exp  |`  RR ) : RR -1-1-onto-> RR+  ->  ( exp  |`  RR ) : RR --> RR+ )
64, 5ax-mp 5 . . . . . 6  |-  ( exp  |`  RR ) : RR --> RR+
7 rpssre 11255 . . . . . 6  |-  RR+  C_  RR
8 fss 5745 . . . . . 6  |-  ( ( ( exp  |`  RR ) : RR --> RR+  /\  RR+  C_  RR )  ->  ( exp  |`  RR ) : RR --> RR )
96, 7, 8mp2an 672 . . . . 5  |-  ( exp  |`  RR ) : RR --> RR
10 iccssre 11631 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
111, 2, 10syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  RR )
12 fssres2 5759 . . . . 5  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  ( A [,] B ) 
C_  RR )  -> 
( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
139, 11, 12sylancr 663 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 ax-resscn 9566 . . . . 5  |-  RR  C_  CC
1511, 14syl6ss 3511 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A [,] B )  C_  CC )
16 efcn 22963 . . . . . 6  |-  exp  e.  ( CC -cn-> CC )
17 rescncf 21526 . . . . . 6  |-  ( ( A [,] B ) 
C_  CC  ->  ( exp 
e.  ( CC -cn-> CC )  ->  ( exp  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> CC ) ) )
1815, 16, 17mpisyl 18 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
19 cncffvrn 21527 . . . . 5  |-  ( ( RR  C_  CC  /\  ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )  ->  ( ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  <-> 
( exp  |`  ( A [,] B ) ) : ( A [,] B ) --> RR ) )
2014, 18, 19sylancr 663 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> RR )  <->  ( exp  |`  ( A [,] B
) ) : ( A [,] B ) --> RR ) )
2113, 20mpbird 232 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
22 reefiso 22968 . . . . . 6  |-  ( exp  |`  RR )  Isom  <  ,  <  ( RR ,  RR+ )
2322a1i 11 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp  |`  RR ) 
Isom  <  ,  <  ( RR ,  RR+ ) )
24 ioossre 11611 . . . . . 6  |-  ( A (,) B )  C_  RR
2524a1i 11 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( A (,) B )  C_  RR )
26 eqidd 2458 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR ) " ( A (,) B ) )  =  ( ( exp  |`  RR ) " ( A (,) B ) ) )
27 isores3 6232 . . . . 5  |-  ( ( ( exp  |`  RR ) 
Isom  <  ,  <  ( RR ,  RR+ )  /\  ( A (,) B ) 
C_  RR  /\  (
( exp  |`  RR )
" ( A (,) B ) )  =  ( ( exp  |`  RR )
" ( A (,) B ) ) )  ->  ( ( exp  |`  RR )  |`  ( A (,) B ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) ) )
2823, 25, 26, 27syl3anc 1228 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR )  |`  ( A (,) B ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) ) )
29 ssid 3518 . . . . . . 7  |-  RR  C_  RR
30 fss 5745 . . . . . . . . 9  |-  ( ( ( exp  |`  RR ) : RR --> RR  /\  RR  C_  CC )  -> 
( exp  |`  RR ) : RR --> CC )
319, 14, 30mp2an 672 . . . . . . . 8  |-  ( exp  |`  RR ) : RR --> CC
32 eqid 2457 . . . . . . . . 9  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3332tgioo2 21433 . . . . . . . . 9  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3432, 33dvres 22440 . . . . . . . 8  |-  ( ( ( RR  C_  CC  /\  ( exp  |`  RR ) : RR --> CC )  /\  ( RR  C_  RR  /\  ( A [,] B )  C_  RR ) )  ->  ( RR  _D  ( ( exp  |`  RR )  |`  ( A [,] B ) ) )  =  ( ( RR  _D  ( exp  |`  RR ) )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
3514, 31, 34mpanl12 682 . . . . . . 7  |-  ( ( RR  C_  RR  /\  ( A [,] B )  C_  RR )  ->  ( RR 
_D  ( ( exp  |`  RR )  |`  ( A [,] B ) ) )  =  ( ( RR  _D  ( exp  |`  RR ) )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) ) ) )
3629, 11, 35sylancr 663 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( ( exp  |`  RR )  |`  ( A [,] B
) ) )  =  ( ( RR  _D  ( exp  |`  RR )
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) ) ) )
3711resabs1d 5313 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  RR )  |`  ( A [,] B ) )  =  ( exp  |`  ( A [,] B ) ) )
3837oveq2d 6312 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( ( exp  |`  RR )  |`  ( A [,] B
) ) )  =  ( RR  _D  ( exp  |`  ( A [,] B ) ) ) )
39 reelprrecn 9601 . . . . . . . . . 10  |-  RR  e.  { RR ,  CC }
40 eff 13828 . . . . . . . . . 10  |-  exp : CC
--> CC
41 ssid 3518 . . . . . . . . . 10  |-  CC  C_  CC
42 dvef 22506 . . . . . . . . . . . . 13  |-  ( CC 
_D  exp )  =  exp
4342dmeqi 5214 . . . . . . . . . . . 12  |-  dom  ( CC  _D  exp )  =  dom  exp
4440fdmi 5742 . . . . . . . . . . . 12  |-  dom  exp  =  CC
4543, 44eqtri 2486 . . . . . . . . . . 11  |-  dom  ( CC  _D  exp )  =  CC
4614, 45sseqtr4i 3532 . . . . . . . . . 10  |-  RR  C_  dom  ( CC  _D  exp )
47 dvres3 22442 . . . . . . . . . 10  |-  ( ( ( RR  e.  { RR ,  CC }  /\  exp : CC --> CC )  /\  ( CC  C_  CC  /\  RR  C_  dom  ( CC  _D  exp )
) )  ->  ( RR  _D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR ) )
4839, 40, 41, 46, 47mp4an 673 . . . . . . . . 9  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( ( CC  _D  exp )  |`  RR )
4942reseq1i 5279 . . . . . . . . 9  |-  ( ( CC  _D  exp )  |`  RR )  =  ( exp  |`  RR )
5048, 49eqtri 2486 . . . . . . . 8  |-  ( RR 
_D  ( exp  |`  RR ) )  =  ( exp  |`  RR )
5150a1i 11 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  RR )
)  =  ( exp  |`  RR ) )
52 iccntr 21451 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
531, 2, 52syl2anc 661 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( int `  ( topGen `  ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5451, 53reseq12d 5284 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( RR 
_D  ( exp  |`  RR ) )  |`  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A [,] B ) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B
) ) )
5536, 38, 543eqtr3d 2506 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  ( A [,] B ) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B ) ) )
56 isoeq1 6216 . . . . 5  |-  ( ( RR  _D  ( exp  |`  ( A [,] B
) ) )  =  ( ( exp  |`  RR )  |`  ( A (,) B
) )  ->  (
( RR  _D  ( exp  |`  ( A [,] B ) ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) )  <-> 
( ( exp  |`  RR )  |`  ( A (,) B
) )  Isom  <  ,  <  ( ( A (,) B ) ,  ( ( exp  |`  RR )
" ( A (,) B ) ) ) ) )
5755, 56syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( RR 
_D  ( exp  |`  ( A [,] B ) ) )  Isom  <  ,  <  ( ( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) )  <-> 
( ( exp  |`  RR )  |`  ( A (,) B
) )  Isom  <  ,  <  ( ( A (,) B ) ,  ( ( exp  |`  RR )
" ( A (,) B ) ) ) ) )
5828, 57mpbird 232 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( RR  _D  ( exp  |`  ( A [,] B ) ) ) 
Isom  <  ,  <  (
( A (,) B
) ,  ( ( exp  |`  RR ) " ( A (,) B ) ) ) )
59 simpr 461 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 (,) 1 ) )
60 eqid 2457 . . 3  |-  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B ) )  =  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) )
611, 2, 3, 21, 58, 59, 60dvcvx 22546 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  ( ( T  x.  ( ( exp  |`  ( A [,] B ) ) `  A ) )  +  ( ( 1  -  T )  x.  (
( exp  |`  ( A [,] B ) ) `
 B ) ) ) )
62 ax-1cn 9567 . . . . . . 7  |-  1  e.  CC
63 ioossre 11611 . . . . . . . . 9  |-  ( 0 (,) 1 )  C_  RR
6463, 59sseldi 3497 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  RR )
6564recnd 9639 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  CC )
66 nncan 9867 . . . . . . 7  |-  ( ( 1  e.  CC  /\  T  e.  CC )  ->  ( 1  -  (
1  -  T ) )  =  T )
6762, 65, 66sylancr 663 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( 1  -  ( 1  -  T
) )  =  T )
6867oveq1d 6311 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( 1  -  ( 1  -  T ) )  x.  A )  =  ( T  x.  A ) )
6968oveq1d 6311 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  A )  +  ( ( 1  -  T )  x.  B
) )  =  ( ( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )
70 ioossicc 11635 . . . . . . 7  |-  ( 0 (,) 1 )  C_  ( 0 [,] 1
)
7170, 59sseldi 3497 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  T  e.  ( 0 [,] 1 ) )
72 iirev 21554 . . . . . 6  |-  ( T  e.  ( 0 [,] 1 )  ->  (
1  -  T )  e.  ( 0 [,] 1 ) )
7371, 72syl 16 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( 1  -  T )  e.  ( 0 [,] 1 ) )
74 lincmb01cmp 11688 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  ( 1  -  T
)  e.  ( 0 [,] 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  A )  +  ( ( 1  -  T )  x.  B
) )  e.  ( A [,] B ) )
7573, 74syldan 470 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( ( 1  -  ( 1  -  T ) )  x.  A )  +  ( ( 1  -  T )  x.  B
) )  e.  ( A [,] B ) )
7669, 75eqeltrrd 2546 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) )  e.  ( A [,] B ) )
77 fvres 5886 . . 3  |-  ( ( ( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) )  e.  ( A [,] B )  ->  (
( exp  |`  ( A [,] B ) ) `
 ( ( T  x.  A )  +  ( ( 1  -  T )  x.  B
) ) )  =  ( exp `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) ) )
7876, 77syl 16 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  =  ( exp `  ( ( T  x.  A )  +  ( ( 1  -  T
)  x.  B ) ) ) )
791rexrd 9660 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  RR* )
802rexrd 9660 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  RR* )
811, 2, 3ltled 9750 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  <_  B
)
82 lbicc2 11661 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
8379, 80, 81, 82syl3anc 1228 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  A  e.  ( A [,] B ) )
84 fvres 5886 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  (
( exp  |`  ( A [,] B ) ) `
 A )  =  ( exp `  A
) )
8583, 84syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  A
)  =  ( exp `  A ) )
8685oveq2d 6312 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( T  x.  ( ( exp  |`  ( A [,] B ) ) `
 A ) )  =  ( T  x.  ( exp `  A ) ) )
87 ubicc2 11662 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
8879, 80, 81, 87syl3anc 1228 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  B  e.  ( A [,] B ) )
89 fvres 5886 . . . . 5  |-  ( B  e.  ( A [,] B )  ->  (
( exp  |`  ( A [,] B ) ) `
 B )  =  ( exp `  B
) )
9088, 89syl 16 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( exp  |`  ( A [,] B
) ) `  B
)  =  ( exp `  B ) )
9190oveq2d 6312 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( 1  -  T )  x.  ( ( exp  |`  ( A [,] B ) ) `
 B ) )  =  ( ( 1  -  T )  x.  ( exp `  B
) ) )
9286, 91oveq12d 6314 . 2  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( ( T  x.  ( ( exp  |`  ( A [,] B
) ) `  A
) )  +  ( ( 1  -  T
)  x.  ( ( exp  |`  ( A [,] B ) ) `  B ) ) )  =  ( ( T  x.  ( exp `  A
) )  +  ( ( 1  -  T
)  x.  ( exp `  B ) ) ) )
9361, 78, 923brtr3d 4485 1  |-  ( ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B )  /\  T  e.  ( 0 (,) 1 ) )  ->  ( exp `  (
( T  x.  A
)  +  ( ( 1  -  T )  x.  B ) ) )  <  ( ( T  x.  ( exp `  A ) )  +  ( ( 1  -  T )  x.  ( exp `  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   {cpr 4034   class class class wbr 4456   dom cdm 5008   ran crn 5009    |` cres 5010   "cima 5011   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594    Isom wiso 5595  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    x. cmul 9514   RR*cxr 9644    < clt 9645    <_ cle 9646    - cmin 9824   RR+crp 11245   (,)cioo 11554   [,]cicc 11557   expce 13808   TopOpenctopn 14838   topGenctg 14854  ℂfldccnfld 18546   intcnt 19644   -cn->ccncf 21505    _D cdv 22392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-fl 11931  df-seq 12110  df-exp 12169  df-fac 12356  df-bc 12383  df-hash 12408  df-shft 12911  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-limsup 13305  df-clim 13322  df-rlim 13323  df-sum 13520  df-ef 13814  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-haus 19942  df-cmp 20013  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-limc 22395  df-dv 22396
This theorem is referenced by: (None)
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