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Theorem dvgt0lem1 19349
Description: Lemma for dvgt0 19351 and dvlt0 19352. (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 10732 . . . . . . 7  |-  ( A [,] B )  C_  RR*
2 simplrl 736 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( A [,] B ) )
31, 2sseldi 3178 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR* )
4 simplrr 737 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( A [,] B ) )
51, 4sseldi 3178 . . . . . 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 10731 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
96, 7, 8syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  RR )
109ad2antrr 706 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( A [,] B
)  C_  RR )
1110, 2sseldd 3181 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR )
1210, 4sseldd 3181 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  RR )
13 simpr 447 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <  Y )
1411, 12, 13ltled 8967 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <_  Y )
15 ubicc2 10753 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  Y  e.  ( X [,] Y
) )
163, 5, 14, 15syl3anc 1182 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( X [,] Y ) )
17 fvres 5542 . . . . 5  |-  ( Y  e.  ( X [,] Y )  ->  (
( F  |`  ( X [,] Y ) ) `
 Y )  =  ( F `  Y
) )
1816, 17syl 15 . . . 4  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( F  |`  ( X [,] Y ) ) `  Y )  =  ( F `  Y ) )
19 lbicc2 10752 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  X  e.  ( X [,] Y
) )
203, 5, 14, 19syl3anc 1182 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( X [,] Y ) )
21 fvres 5542 . . . . 5  |-  ( X  e.  ( X [,] Y )  ->  (
( F  |`  ( X [,] Y ) ) `
 X )  =  ( F `  X
) )
2220, 21syl 15 . . . 4  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( F  |`  ( X [,] Y ) ) `  X )  =  ( F `  X ) )
2318, 22oveq12d 5876 . . 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 5873 . 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 10720 . . . . . 6  |-  ( ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) )  -> 
( X [,] Y
)  C_  ( A [,] B ) )
2625ad2antlr 707 . . . . 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 706 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
29 rescncf 18401 . . . . 5  |-  ( ( X [,] Y ) 
C_  ( A [,] B )  ->  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( X [,] Y
) )  e.  ( ( X [,] Y
) -cn-> RR ) ) )
3026, 28, 29sylc 56 . . . 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 706 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  F
) : ( A (,) B ) --> S )
336ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  e.  RR )
3433rexrd 8881 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  e.  RR* )
357ad2antrr 706 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR )
36 elicc2 10715 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
3733, 35, 36syl2anc 642 . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) )
3938simp2d 968 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  <_  X )
40 iooss1 10691 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  <_  X )  ->  ( X (,) Y )  C_  ( A (,) Y ) )
4134, 39, 40syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) Y ) )
4235rexrd 8881 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR* )
43 elicc2 10715 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Y  e.  ( A [,] B )  <-> 
( Y  e.  RR  /\  A  <_  Y  /\  Y  <_  B ) ) )
4433, 35, 43syl2anc 642 . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( Y  e.  RR  /\  A  <_  Y  /\  Y  <_  B ) )
4645simp3d 969 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  <_  B )
47 iooss2 10692 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  Y  <_  B )  ->  ( A (,) Y )  C_  ( A (,) B ) )
4842, 46, 47syl2anc 642 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( A (,) Y
)  C_  ( A (,) B ) )
4941, 48sstrd 3189 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) B ) )
50 fssres 5408 . . . . . . 7  |-  ( ( ( RR  _D  F
) : ( A (,) B ) --> S  /\  ( X (,) Y )  C_  ( A (,) B ) )  ->  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S )
5132, 49, 50syl2anc 642 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( RR  _D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S )
52 ax-resscn 8794 . . . . . . . . . 10  |-  RR  C_  CC
5352a1i 10 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  RR  C_  CC )
54 cncff 18397 . . . . . . . . . . . 12  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
5527, 54syl 15 . . . . . . . . . . 11  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5655ad2antrr 706 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> RR )
57 fss 5397 . . . . . . . . . 10  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
5856, 52, 57sylancl 643 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> CC )
59 iccssre 10731 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
6011, 12, 59syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X [,] Y
)  C_  RR )
61 eqid 2283 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6261tgioo2 18309 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
6361, 62dvres 19261 . . . . . . . . 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 1183 . . . . . . . 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 18326 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6611, 12, 65syl2anc 642 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6766reseq2d 4955 . . . . . . . 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 2315 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) )
6968feq1d 5379 . . . . . 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 223 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) ) : ( X (,) Y ) --> S )
71 fdm 5393 . . . . 5  |-  ( ( RR  _D  ( F  |`  ( X [,] Y
) ) ) : ( X (,) Y
) --> S  ->  dom  ( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( X (,) Y ) )
7270, 71syl 15 . . . 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 19339 . . 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 ) ) )
74 ffvelrn 5663 . . . . . 6  |-  ( ( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) : ( X (,) Y ) --> S  /\  z  e.  ( X (,) Y ) )  -> 
( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) `  z )  e.  S )
7570, 74sylan 457 . . . . 5  |-  ( ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) ) )  /\  X  <  Y
)  /\  z  e.  ( X (,) Y ) )  ->  ( ( RR  _D  ( F  |`  ( X [,] Y ) ) ) `  z
)  e.  S )
76 eleq1 2343 . . . . 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 ) )
7775, 76syl5ibcom 211 . . . 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 ) )
7877rexlimdva 2667 . . 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 ) )
7973, 78mpd 14 . 2  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( ( ( F  |`  ( X [,] Y ) ) `  Y )  -  (
( F  |`  ( X [,] Y ) ) `
 X ) )  /  ( Y  -  X ) )  e.  S )
8024, 79eqeltrrd 2358 1  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( ( F `
 Y )  -  ( F `  X ) )  /  ( Y  -  X ) )  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   E.wrex 2544    C_ wss 3152   class class class wbr 4023   dom cdm 4689   ran crn 4690    |` cres 4691   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   RR*cxr 8866    < clt 8867    <_ cle 8868    - cmin 9037    / cdiv 9423   (,)cioo 10656   [,]cicc 10659   TopOpenctopn 13326   topGenctg 13342  ℂfldccnfld 16377   intcnt 16754   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  dvgt0  19351  dvlt0  19352  dvge0  19353
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
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