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Theorem dvgt0lem1 21592
Description: Lemma for dvgt0 21594 and dvlt0 21595. (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 11481 . . . . . . 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 3454 . . . . . 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 3454 . . . . . 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 11480 . . . . . . . . . 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 3457 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR )
1210, 4sseldd 3457 . . . . . . 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 9625 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <_  Y )
15 ubicc2 11505 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  Y  e.  ( X [,] Y
) )
163, 5, 14, 15syl3anc 1219 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( X [,] Y ) )
17 fvres 5805 . . . . 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 11504 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  X  e.  ( X [,] Y
) )
203, 5, 14, 19syl3anc 1219 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( X [,] Y ) )
21 fvres 5805 . . . . 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 6210 . . 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 6207 . 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 11469 . . . . . 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 20591 . . . . 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 9536 . . . . . . . . 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 11463 . . . . . . . . . . . 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 11438 . . . . . . . . 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 9536 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR* )
43 elicc2 11463 . . . . . . . . . . . 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 11439 . . . . . . . . 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 3466 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) B ) )
50 fssres 5678 . . . . . . 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 9442 . . . . . . . . . 10  |-  RR  C_  CC
5352a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  RR  C_  CC )
54 cncff 20587 . . . . . . . . . . . 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 5667 . . . . . . . . . 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 11480 . . . . . . . . . 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 2451 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6261tgioo2 20498 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
6361, 62dvres 21504 . . . . . . . . 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 1220 . . . . . . . 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 20516 . . . . . . . . . 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 5210 . . . . . . . 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 2492 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) )
6968feq1d 5646 . . . . . 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 5663 . . . . 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 21582 . . 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 5944 . . . . 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 2523 . . . . 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 2939 . . 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 2540 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 1370    e. wcel 1758   E.wrex 2796    C_ wss 3428   class class class wbr 4392   dom cdm 4940   ran crn 4941    |` cres 4942   -->wf 5514   ` cfv 5518  (class class class)co 6192   CCcc 9383   RRcr 9384   RR*cxr 9520    < clt 9521    <_ cle 9522    - cmin 9698    / cdiv 10096   (,)cioo 11403   [,]cicc 11406   TopOpenctopn 14464   topGenctg 14480  ℂfldccnfld 17929   intcnt 18739   -cn->ccncf 20570    _D cdv 21456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-inf2 7950  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462  ax-pre-sup 9463  ax-addf 9464  ax-mulf 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-int 4229  df-iun 4273  df-iin 4274  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-se 4780  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-isom 5527  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-of 6422  df-om 6579  df-1st 6679  df-2nd 6680  df-supp 6793  df-recs 6934  df-rdg 6968  df-1o 7022  df-2o 7023  df-oadd 7026  df-er 7203  df-map 7318  df-pm 7319  df-ixp 7366  df-en 7413  df-dom 7414  df-sdom 7415  df-fin 7416  df-fsupp 7724  df-fi 7764  df-sup 7794  df-oi 7827  df-card 8212  df-cda 8440  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-div 10097  df-nn 10426  df-2 10483  df-3 10484  df-4 10485  df-5 10486  df-6 10487  df-7 10488  df-8 10489  df-9 10490  df-10 10491  df-n0 10683  df-z 10750  df-dec 10859  df-uz 10965  df-q 11057  df-rp 11095  df-xneg 11192  df-xadd 11193  df-xmul 11194  df-ioo 11407  df-ico 11409  df-icc 11410  df-fz 11541  df-fzo 11652  df-seq 11910  df-exp 11969  df-hash 12207  df-cj 12692  df-re 12693  df-im 12694  df-sqr 12828  df-abs 12829  df-struct 14280  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-ress 14285  df-plusg 14355  df-mulr 14356  df-starv 14357  df-sca 14358  df-vsca 14359  df-ip 14360  df-tset 14361  df-ple 14362  df-ds 14364  df-unif 14365  df-hom 14366  df-cco 14367  df-rest 14465  df-topn 14466  df-0g 14484  df-gsum 14485  df-topgen 14486  df-pt 14487  df-prds 14490  df-xrs 14544  df-qtop 14549  df-imas 14550  df-xps 14552  df-mre 14628  df-mrc 14629  df-acs 14631  df-mnd 15519  df-submnd 15569  df-mulg 15652  df-cntz 15939  df-cmn 16385  df-psmet 17920  df-xmet 17921  df-met 17922  df-bl 17923  df-mopn 17924  df-fbas 17925  df-fg 17926  df-cnfld 17930  df-top 18621  df-bases 18623  df-topon 18624  df-topsp 18625  df-cld 18741  df-ntr 18742  df-cls 18743  df-nei 18820  df-lp 18858  df-perf 18859  df-cn 18949  df-cnp 18950  df-haus 19037  df-cmp 19108  df-tx 19253  df-hmeo 19446  df-fil 19537  df-fm 19629  df-flim 19630  df-flf 19631  df-xms 20013  df-ms 20014  df-tms 20015  df-cncf 20572  df-limc 21459  df-dv 21460
This theorem is referenced by:  dvgt0  21594  dvlt0  21595  dvge0  21596
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