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Theorem dvgt0lem1 22273
Description: Lemma for dvgt0 22275 and dvlt0 22276. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
dvgt0.a  |-  ( ph  ->  A  e.  RR )
dvgt0.b  |-  ( ph  ->  B  e.  RR )
dvgt0.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
dvgt0lem.d  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> S )
Assertion
Ref Expression
dvgt0lem1  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( ( F `
 Y )  -  ( F `  X ) )  /  ( Y  -  X ) )  e.  S )

Proof of Theorem dvgt0lem1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 iccssxr 11613 . . . . . . 7  |-  ( A [,] B )  C_  RR*
2 simplrl 759 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( A [,] B ) )
31, 2sseldi 3485 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR* )
4 simplrr 760 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( A [,] B ) )
51, 4sseldi 3485 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  RR* )
6 dvgt0.a . . . . . . . . . 10  |-  ( ph  ->  A  e.  RR )
7 dvgt0.b . . . . . . . . . 10  |-  ( ph  ->  B  e.  RR )
8 iccssre 11612 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
96, 7, 8syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  RR )
109ad2antrr 725 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( A [,] B
)  C_  RR )
1110, 2sseldd 3488 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR )
1210, 4sseldd 3488 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  RR )
13 simpr 461 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <  Y )
1411, 12, 13ltled 9733 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <_  Y )
15 ubicc2 11643 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  Y  e.  ( X [,] Y
) )
163, 5, 14, 15syl3anc 1227 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( X [,] Y ) )
17 fvres 5867 . . . . 5  |-  ( Y  e.  ( X [,] Y )  ->  (
( F  |`  ( X [,] Y ) ) `
 Y )  =  ( F `  Y
) )
1816, 17syl 16 . . . 4  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( F  |`  ( X [,] Y ) ) `  Y )  =  ( F `  Y ) )
19 lbicc2 11642 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  X  e.  ( X [,] Y
) )
203, 5, 14, 19syl3anc 1227 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( X [,] Y ) )
21 fvres 5867 . . . . 5  |-  ( X  e.  ( X [,] Y )  ->  (
( F  |`  ( X [,] Y ) ) `
 X )  =  ( F `  X
) )
2220, 21syl 16 . . . 4  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( F  |`  ( X [,] Y ) ) `  X )  =  ( F `  X ) )
2318, 22oveq12d 6296 . . 3  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( ( F  |`  ( X [,] Y
) ) `  Y
)  -  ( ( F  |`  ( X [,] Y ) ) `  X ) )  =  ( ( F `  Y )  -  ( F `  X )
) )
2423oveq1d 6293 . 2  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  (
( F  |`  ( X [,] Y ) ) `
 X ) )  /  ( Y  -  X ) )  =  ( ( ( F `
 Y )  -  ( F `  X ) )  /  ( Y  -  X ) ) )
25 iccss2 11601 . . . . . 6  |-  ( ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) )  -> 
( X [,] Y
)  C_  ( A [,] B ) )
2625ad2antlr 726 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X [,] Y
)  C_  ( A [,] B ) )
27 dvgt0.f . . . . . 6  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
2827ad2antrr 725 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
29 rescncf 21271 . . . . 5  |-  ( ( X [,] Y ) 
C_  ( A [,] B )  ->  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( X [,] Y
) )  e.  ( ( X [,] Y
) -cn-> RR ) ) )
3026, 28, 29sylc 60 . . . 4  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( F  |`  ( X [,] Y ) )  e.  ( ( X [,] Y ) -cn-> RR ) )
31 dvgt0lem.d . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> S )
3231ad2antrr 725 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  F
) : ( A (,) B ) --> S )
336ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  e.  RR )
3433rexrd 9643 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  e.  RR* )
357ad2antrr 725 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR )
36 elicc2 11595 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
3733, 35, 36syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
382, 37mpbid 210 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) )
3938simp2d 1008 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  <_  X )
40 iooss1 11570 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  <_  X )  ->  ( X (,) Y )  C_  ( A (,) Y ) )
4134, 39, 40syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) Y ) )
4235rexrd 9643 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR* )
43 elicc2 11595 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Y  e.  ( A [,] B )  <-> 
( Y  e.  RR  /\  A  <_  Y  /\  Y  <_  B ) ) )
4433, 35, 43syl2anc 661 . . . . . . . . . . 11  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( Y  e.  ( A [,] B )  <-> 
( Y  e.  RR  /\  A  <_  Y  /\  Y  <_  B ) ) )
454, 44mpbid 210 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( Y  e.  RR  /\  A  <_  Y  /\  Y  <_  B ) )
4645simp3d 1009 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  <_  B )
47 iooss2 11571 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  Y  <_  B )  ->  ( A (,) Y )  C_  ( A (,) B ) )
4842, 46, 47syl2anc 661 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( A (,) Y
)  C_  ( A (,) B ) )
4941, 48sstrd 3497 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) B ) )
5032, 49fssresd 5739 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( RR  _D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S )
51 ax-resscn 9549 . . . . . . . . . 10  |-  RR  C_  CC
5251a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  RR  C_  CC )
53 cncff 21267 . . . . . . . . . . . 12  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
5427, 53syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5554ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> RR )
56 fss 5726 . . . . . . . . . 10  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
5755, 51, 56sylancl 662 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> CC )
58 iccssre 11612 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
5911, 12, 58syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X [,] Y
)  C_  RR )
60 eqid 2441 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6160tgioo2 21178 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
6260, 61dvres 22185 . . . . . . . . 9  |-  ( ( ( RR  C_  CC  /\  F : ( A [,] B ) --> CC )  /\  ( ( A [,] B ) 
C_  RR  /\  ( X [,] Y )  C_  RR ) )  ->  ( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( ( RR  _D  F
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( X [,] Y ) ) ) )
6352, 57, 10, 59, 62syl22anc 1228 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) ) ) )
64 iccntr 21196 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6511, 12, 64syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6665reseq2d 5260 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) ) )  =  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) )
6763, 66eqtrd 2482 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) )
6867feq1d 5704 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) : ( X (,) Y ) --> S  <-> 
( ( RR  _D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S ) )
6950, 68mpbird 232 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) ) : ( X (,) Y ) --> S )
70 fdm 5722 . . . . 5  |-  ( ( RR  _D  ( F  |`  ( X [,] Y
) ) ) : ( X (,) Y
) --> S  ->  dom  ( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( X (,) Y ) )
7169, 70syl 16 . . . 4  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  dom  ( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( X (,) Y ) )
7211, 12, 13, 30, 71mvth 22263 . . 3  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  E. z  e.  ( X (,) Y ) ( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) `
 z )  =  ( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  (
( F  |`  ( X [,] Y ) ) `
 X ) )  /  ( Y  -  X ) ) )
7369ffvelrnda 6013 . . . . 5  |-  ( ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) ) )  /\  X  <  Y
)  /\  z  e.  ( X (,) Y ) )  ->  ( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) `  z
)  e.  S )
74 eleq1 2513 . . . . 5  |-  ( ( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) `
 z )  =  ( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  (
( F  |`  ( X [,] Y ) ) `
 X ) )  /  ( Y  -  X ) )  -> 
( ( ( RR 
_D  ( F  |`  ( X [,] Y ) ) ) `  z
)  e.  S  <->  ( (
( ( F  |`  ( X [,] Y ) ) `  Y )  -  ( ( F  |`  ( X [,] Y
) ) `  X
) )  /  ( Y  -  X )
)  e.  S ) )
7573, 74syl5ibcom 220 . . . 4  |-  ( ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) ) )  /\  X  <  Y
)  /\  z  e.  ( X (,) Y ) )  ->  ( (
( RR  _D  ( F  |`  ( X [,] Y ) ) ) `
 z )  =  ( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  (
( F  |`  ( X [,] Y ) ) `
 X ) )  /  ( Y  -  X ) )  -> 
( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  (
( F  |`  ( X [,] Y ) ) `
 X ) )  /  ( Y  -  X ) )  e.  S ) )
7675rexlimdva 2933 . . 3  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( E. z  e.  ( X (,) Y
) ( ( RR 
_D  ( F  |`  ( X [,] Y ) ) ) `  z
)  =  ( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  ( ( F  |`  ( X [,] Y
) ) `  X
) )  /  ( Y  -  X )
)  ->  ( (
( ( F  |`  ( X [,] Y ) ) `  Y )  -  ( ( F  |`  ( X [,] Y
) ) `  X
) )  /  ( Y  -  X )
)  e.  S ) )
7772, 76mpd 15 . 2  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  (
( F  |`  ( X [,] Y ) ) `
 X ) )  /  ( Y  -  X ) )  e.  S )
7824, 77eqeltrrd 2530 1  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( ( F `
 Y )  -  ( F `  X ) )  /  ( Y  -  X ) )  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   E.wrex 2792    C_ wss 3459   class class class wbr 4434   dom cdm 4986   ran crn 4987    |` cres 4988   -->wf 5571   ` cfv 5575  (class class class)co 6278   CCcc 9490   RRcr 9491   RR*cxr 9627    < clt 9628    <_ cle 9629    - cmin 9807    / cdiv 10209   (,)cioo 11535   [,]cicc 11538   TopOpenctopn 14693   topGenctg 14709  ℂfldccnfld 18291   intcnt 19388   -cn->ccncf 21250    _D cdv 22137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4545  ax-sep 4555  ax-nul 4563  ax-pow 4612  ax-pr 4673  ax-un 6574  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571  ax-mulf 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3419  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-pss 3475  df-nul 3769  df-if 3924  df-pw 3996  df-sn 4012  df-pr 4014  df-tp 4016  df-op 4018  df-uni 4232  df-int 4269  df-iun 4314  df-iin 4315  df-br 4435  df-opab 4493  df-mpt 4494  df-tr 4528  df-eprel 4778  df-id 4782  df-po 4787  df-so 4788  df-fr 4825  df-se 4826  df-we 4827  df-ord 4868  df-on 4869  df-lim 4870  df-suc 4871  df-xp 4992  df-rel 4993  df-cnv 4994  df-co 4995  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fun 5577  df-fn 5578  df-f 5579  df-f1 5580  df-fo 5581  df-f1o 5582  df-fv 5583  df-isom 5584  df-riota 6239  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6522  df-om 6683  df-1st 6782  df-2nd 6783  df-supp 6901  df-recs 7041  df-rdg 7075  df-1o 7129  df-2o 7130  df-oadd 7133  df-er 7310  df-map 7421  df-pm 7422  df-ixp 7469  df-en 7516  df-dom 7517  df-sdom 7518  df-fin 7519  df-fsupp 7829  df-fi 7870  df-sup 7900  df-oi 7935  df-card 8320  df-cda 8548  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9809  df-neg 9810  df-div 10210  df-nn 10540  df-2 10597  df-3 10598  df-4 10599  df-5 10600  df-6 10601  df-7 10602  df-8 10603  df-9 10604  df-10 10605  df-n0 10799  df-z 10868  df-dec 10982  df-uz 11088  df-q 11189  df-rp 11227  df-xneg 11324  df-xadd 11325  df-xmul 11326  df-ioo 11539  df-ico 11541  df-icc 11542  df-fz 11679  df-fzo 11801  df-seq 12084  df-exp 12143  df-hash 12382  df-cj 12908  df-re 12909  df-im 12910  df-sqrt 13044  df-abs 13045  df-struct 14508  df-ndx 14509  df-slot 14510  df-base 14511  df-sets 14512  df-ress 14513  df-plusg 14584  df-mulr 14585  df-starv 14586  df-sca 14587  df-vsca 14588  df-ip 14589  df-tset 14590  df-ple 14591  df-ds 14593  df-unif 14594  df-hom 14595  df-cco 14596  df-rest 14694  df-topn 14695  df-0g 14713  df-gsum 14714  df-topgen 14715  df-pt 14716  df-prds 14719  df-xrs 14773  df-qtop 14778  df-imas 14779  df-xps 14781  df-mre 14857  df-mrc 14858  df-acs 14860  df-mgm 15743  df-sgrp 15782  df-mnd 15792  df-submnd 15838  df-mulg 15931  df-cntz 16226  df-cmn 16671  df-psmet 18282  df-xmet 18283  df-met 18284  df-bl 18285  df-mopn 18286  df-fbas 18287  df-fg 18288  df-cnfld 18292  df-top 19269  df-bases 19271  df-topon 19272  df-topsp 19273  df-cld 19390  df-ntr 19391  df-cls 19392  df-nei 19469  df-lp 19507  df-perf 19508  df-cn 19598  df-cnp 19599  df-haus 19686  df-cmp 19757  df-tx 19933  df-hmeo 20126  df-fil 20217  df-fm 20309  df-flim 20310  df-flf 20311  df-xms 20693  df-ms 20694  df-tms 20695  df-cncf 21252  df-limc 22140  df-dv 22141
This theorem is referenced by:  dvgt0  22275  dvlt0  22276  dvge0  22277
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