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Theorem dvgt0lem1 21454
Description: Lemma for dvgt0 21456 and dvlt0 21457. (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 11370 . . . . . . 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 3349 . . . . . 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 3349 . . . . . 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 11369 . . . . . . . . . 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 3352 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR )
1210, 4sseldd 3352 . . . . . . 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 9514 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <_  Y )
15 ubicc2 11394 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  Y  e.  ( X [,] Y
) )
163, 5, 14, 15syl3anc 1218 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( X [,] Y ) )
17 fvres 5699 . . . . 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 11393 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  X  e.  ( X [,] Y
) )
203, 5, 14, 19syl3anc 1218 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( X [,] Y ) )
21 fvres 5699 . . . . 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 6104 . . 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 6101 . 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 11358 . . . . . 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 20453 . . . . 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 9425 . . . . . . . . 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 11352 . . . . . . . . . . . 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 1001 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  <_  X )
40 iooss1 11327 . . . . . . . . 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 9425 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR* )
43 elicc2 11352 . . . . . . . . . . . 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 1002 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  <_  B )
47 iooss2 11328 . . . . . . . . 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 3361 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) B ) )
50 fssres 5573 . . . . . . 7  |-  ( ( ( RR  _D  F
) : ( A (,) B ) --> S  /\  ( X (,) Y )  C_  ( A (,) B ) )  ->  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S )
5132, 49, 50syl2anc 661 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( RR  _D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S )
52 ax-resscn 9331 . . . . . . . . . 10  |-  RR  C_  CC
5352a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  RR  C_  CC )
54 cncff 20449 . . . . . . . . . . . 12  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
5527, 54syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5655ad2antrr 725 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> RR )
57 fss 5562 . . . . . . . . . 10  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
5856, 52, 57sylancl 662 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> CC )
59 iccssre 11369 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
6011, 12, 59syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X [,] Y
)  C_  RR )
61 eqid 2438 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6261tgioo2 20360 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
6361, 62dvres 21366 . . . . . . . . 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 ) ) ) )
6453, 58, 10, 60, 63syl22anc 1219 . . . . . . . 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 ) ) ) )
65 iccntr 20378 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6611, 12, 65syl2anc 661 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6766reseq2d 5105 . . . . . . . 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 ) ) )
6864, 67eqtrd 2470 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) )
6968feq1d 5541 . . . . . 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 ) )
7051, 69mpbird 232 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) ) : ( X (,) Y ) --> S )
71 fdm 5558 . . . . 5  |-  ( ( RR  _D  ( F  |`  ( X [,] Y
) ) ) : ( X (,) Y
) --> S  ->  dom  ( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( X (,) Y ) )
7270, 71syl 16 . . . 4  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  dom  ( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( X (,) Y ) )
7311, 12, 13, 30, 72mvth 21444 . . 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 ) ) )
7470ffvelrnda 5838 . . . . 5  |-  ( ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) ) )  /\  X  <  Y
)  /\  z  e.  ( X (,) Y ) )  ->  ( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) `  z
)  e.  S )
75 eleq1 2498 . . . . 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 ) )
7674, 75syl5ibcom 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 ) )
7776rexlimdva 2836 . . 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 ) )
7873, 77mpd 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 )
7924, 78eqeltrrd 2513 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 965    = wceq 1369    e. wcel 1756   E.wrex 2711    C_ wss 3323   class class class wbr 4287   dom cdm 4835   ran crn 4836    |` cres 4837   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   RRcr 9273   RR*cxr 9409    < clt 9410    <_ cle 9411    - cmin 9587    / cdiv 9985   (,)cioo 11292   [,]cicc 11295   TopOpenctopn 14352   topGenctg 14368  ℂfldccnfld 17798   intcnt 18601   -cn->ccncf 20432    _D cdv 21318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-iin 4169  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-supp 6686  df-recs 6824  df-rdg 6858  df-1o 6912  df-2o 6913  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-ixp 7256  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fsupp 7613  df-fi 7653  df-sup 7683  df-oi 7716  df-card 8101  df-cda 8329  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-ioo 11296  df-ico 11298  df-icc 11299  df-fz 11430  df-fzo 11541  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-sca 14246  df-vsca 14247  df-ip 14248  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-hom 14254  df-cco 14255  df-rest 14353  df-topn 14354  df-0g 14372  df-gsum 14373  df-topgen 14374  df-pt 14375  df-prds 14378  df-xrs 14432  df-qtop 14437  df-imas 14438  df-xps 14440  df-mre 14516  df-mrc 14517  df-acs 14519  df-mnd 15407  df-submnd 15457  df-mulg 15539  df-cntz 15826  df-cmn 16270  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-fbas 17794  df-fg 17795  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cld 18603  df-ntr 18604  df-cls 18605  df-nei 18682  df-lp 18720  df-perf 18721  df-cn 18811  df-cnp 18812  df-haus 18899  df-cmp 18970  df-tx 19115  df-hmeo 19308  df-fil 19399  df-fm 19491  df-flim 19492  df-flf 19493  df-xms 19875  df-ms 19876  df-tms 19877  df-cncf 20434  df-limc 21321  df-dv 21322
This theorem is referenced by:  dvgt0  21456  dvlt0  21457  dvge0  21458
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