MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvgt0lem1 Structured version   Unicode version

Theorem dvgt0lem1 19887
Description: Lemma for dvgt0 19889 and dvlt0 19890. (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 10994 . . . . . . 7  |-  ( A [,] B )  C_  RR*
2 simplrl 738 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( A [,] B ) )
31, 2sseldi 3347 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR* )
4 simplrr 739 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( A [,] B ) )
51, 4sseldi 3347 . . . . . 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 10993 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
96, 7, 8syl2anc 644 . . . . . . . . 9  |-  ( ph  ->  ( A [,] B
)  C_  RR )
109ad2antrr 708 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( A [,] B
)  C_  RR )
1110, 2sseldd 3350 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  RR )
1210, 4sseldd 3350 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  RR )
13 simpr 449 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <  Y )
1411, 12, 13ltled 9222 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  <_  Y )
15 ubicc2 11015 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  Y  e.  ( X [,] Y
) )
163, 5, 14, 15syl3anc 1185 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  e.  ( X [,] Y ) )
17 fvres 5746 . . . . 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 11014 . . . . . 6  |-  ( ( X  e.  RR*  /\  Y  e.  RR*  /\  X  <_  Y )  ->  X  e.  ( X [,] Y
) )
203, 5, 14, 19syl3anc 1185 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  X  e.  ( X [,] Y ) )
21 fvres 5746 . . . . 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 6100 . . 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 6097 . 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 10982 . . . . . 6  |-  ( ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B ) )  -> 
( X [,] Y
)  C_  ( A [,] B ) )
2625ad2antlr 709 . . . . 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 708 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
29 rescncf 18928 . . . . 5  |-  ( ( X [,] Y ) 
C_  ( A [,] B )  ->  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( X [,] Y
) )  e.  ( ( X [,] Y
) -cn-> RR ) ) )
3026, 28, 29sylc 59 . . . 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 708 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  F
) : ( A (,) B ) --> S )
336ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  e.  RR )
3433rexrd 9135 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  e.  RR* )
357ad2antrr 708 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR )
36 elicc2 10976 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( X  e.  ( A [,] B )  <-> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) ) )
3733, 35, 36syl2anc 644 . . . . . . . . . . 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 203 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X  e.  RR  /\  A  <_  X  /\  X  <_  B ) )
3938simp2d 971 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  A  <_  X )
40 iooss1 10952 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  A  <_  X )  ->  ( X (,) Y )  C_  ( A (,) Y ) )
4134, 39, 40syl2anc 644 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) Y ) )
4235rexrd 9135 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  B  e.  RR* )
43 elicc2 10976 . . . . . . . . . . . 12  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( Y  e.  ( A [,] B )  <-> 
( Y  e.  RR  /\  A  <_  Y  /\  Y  <_  B ) ) )
4433, 35, 43syl2anc 644 . . . . . . . . . . 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 203 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( Y  e.  RR  /\  A  <_  Y  /\  Y  <_  B ) )
4645simp3d 972 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  Y  <_  B )
47 iooss2 10953 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  Y  <_  B )  ->  ( A (,) Y )  C_  ( A (,) B ) )
4842, 46, 47syl2anc 644 . . . . . . . 8  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( A (,) Y
)  C_  ( A (,) B ) )
4941, 48sstrd 3359 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X (,) Y
)  C_  ( A (,) B ) )
50 fssres 5611 . . . . . . 7  |-  ( ( ( RR  _D  F
) : ( A (,) B ) --> S  /\  ( X (,) Y )  C_  ( A (,) B ) )  ->  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S )
5132, 49, 50syl2anc 644 . . . . . 6  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( RR  _D  F )  |`  ( X (,) Y ) ) : ( X (,) Y ) --> S )
52 ax-resscn 9048 . . . . . . . . . 10  |-  RR  C_  CC
5352a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  RR  C_  CC )
54 cncff 18924 . . . . . . . . . . . 12  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
5527, 54syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F : ( A [,] B ) --> RR )
5655ad2antrr 708 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> RR )
57 fss 5600 . . . . . . . . . 10  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
5856, 52, 57sylancl 645 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  ->  F : ( A [,] B ) --> CC )
59 iccssre 10993 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( X [,] Y
)  C_  RR )
6011, 12, 59syl2anc 644 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( X [,] Y
)  C_  RR )
61 eqid 2437 . . . . . . . . . 10  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
6261tgioo2 18835 . . . . . . . . . 10  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
6361, 62dvres 19799 . . . . . . . . 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 1186 . . . . . . . 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 18853 . . . . . . . . . 10  |-  ( ( X  e.  RR  /\  Y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6611, 12, 65syl2anc 644 . . . . . . . . 9  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( ( int `  ( topGen `
 ran  (,) )
) `  ( X [,] Y ) )  =  ( X (,) Y
) )
6766reseq2d 5147 . . . . . . . 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 2469 . . . . . . 7  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) )  =  ( ( RR 
_D  F )  |`  ( X (,) Y ) ) )
6968feq1d 5581 . . . . . 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 225 . . . . 5  |-  ( ( ( ph  /\  ( X  e.  ( A [,] B )  /\  Y  e.  ( A [,] B
) ) )  /\  X  <  Y )  -> 
( RR  _D  ( F  |`  ( X [,] Y ) ) ) : ( X (,) Y ) --> S )
71 fdm 5596 . . . . 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 19877 . . 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 5871 . . . . 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 2497 . . . . 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 213 . . . 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 2831 . . 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 2512 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 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   E.wrex 2707    C_ wss 3321   class class class wbr 4213   dom cdm 4879   ran crn 4880    |` cres 4881   -->wf 5451   ` cfv 5455  (class class class)co 6082   CCcc 8989   RRcr 8990   RR*cxr 9120    < clt 9121    <_ cle 9122    - cmin 9292    / cdiv 9678   (,)cioo 10917   [,]cicc 10920   TopOpenctopn 13650   topGenctg 13666  ℂfldccnfld 16704   intcnt 17082   -cn->ccncf 18907    _D cdv 19751
This theorem is referenced by:  dvgt0  19889  dvlt0  19890  dvge0  19891
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069  ax-addf 9070  ax-mulf 9071
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-iin 4097  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-se 4543  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-isom 5464  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-of 6306  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-1o 6725  df-2o 6726  df-oadd 6729  df-er 6906  df-map 7021  df-pm 7022  df-ixp 7065  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-fi 7417  df-sup 7447  df-oi 7480  df-card 7827  df-cda 8049  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-2 10059  df-3 10060  df-4 10061  df-5 10062  df-6 10063  df-7 10064  df-8 10065  df-9 10066  df-10 10067  df-n0 10223  df-z 10284  df-dec 10384  df-uz 10490  df-q 10576  df-rp 10614  df-xneg 10711  df-xadd 10712  df-xmul 10713  df-ioo 10921  df-ico 10923  df-icc 10924  df-fz 11045  df-fzo 11137  df-seq 11325  df-exp 11384  df-hash 11620  df-cj 11905  df-re 11906  df-im 11907  df-sqr 12041  df-abs 12042  df-struct 13472  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-ress 13477  df-plusg 13543  df-mulr 13544  df-starv 13545  df-sca 13546  df-vsca 13547  df-tset 13549  df-ple 13550  df-ds 13552  df-unif 13553  df-hom 13554  df-cco 13555  df-rest 13651  df-topn 13652  df-topgen 13668  df-pt 13669  df-prds 13672  df-xrs 13727  df-0g 13728  df-gsum 13729  df-qtop 13734  df-imas 13735  df-xps 13737  df-mre 13812  df-mrc 13813  df-acs 13815  df-mnd 14691  df-submnd 14740  df-mulg 14816  df-cntz 15117  df-cmn 15415  df-psmet 16695  df-xmet 16696  df-met 16697  df-bl 16698  df-mopn 16699  df-fbas 16700  df-fg 16701  df-cnfld 16705  df-top 16964  df-bases 16966  df-topon 16967  df-topsp 16968  df-cld 17084  df-ntr 17085  df-cls 17086  df-nei 17163  df-lp 17201  df-perf 17202  df-cn 17292  df-cnp 17293  df-haus 17380  df-cmp 17451  df-tx 17595  df-hmeo 17788  df-fil 17879  df-fm 17971  df-flim 17972  df-flf 17973  df-xms 18351  df-ms 18352  df-tms 18353  df-cncf 18909  df-limc 19754  df-dv 19755
  Copyright terms: Public domain W3C validator