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Theorem dvbdfbdioolem1 37375
Description: Given a function with bounded derivative, on an open interval, here is an absolute bound to the difference of the image of two points in the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypotheses
Ref Expression
dvbdfbdioolem1.a  |-  ( ph  ->  A  e.  RR )
dvbdfbdioolem1.b  |-  ( ph  ->  B  e.  RR )
dvbdfbdioolem1.f  |-  ( ph  ->  F : ( A (,) B ) --> RR )
dvbdfbdioolem1.dmdv  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
dvbdfbdioolem1.k  |-  ( ph  ->  K  e.  RR )
dvbdfbdioolem1.dvbd  |-  ( ph  ->  A. x  e.  ( A (,) B ) ( abs `  (
( RR  _D  F
) `  x )
)  <_  K )
dvbdfbdioolem1.c  |-  ( ph  ->  C  e.  ( A (,) B ) )
dvbdfbdioolem1.d  |-  ( ph  ->  D  e.  ( C (,) B ) )
Assertion
Ref Expression
dvbdfbdioolem1  |-  ( ph  ->  ( ( abs `  (
( F `  D
)  -  ( F `
 C ) ) )  <_  ( K  x.  ( D  -  C
) )  /\  ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A ) ) ) )
Distinct variable groups:    x, A    x, B    x, C    x, D    x, F    x, K    ph, x

Proof of Theorem dvbdfbdioolem1
StepHypRef Expression
1 ioossre 11696 . . . 4  |-  ( A (,) B )  C_  RR
2 dvbdfbdioolem1.c . . . 4  |-  ( ph  ->  C  e.  ( A (,) B ) )
31, 2sseldi 3468 . . 3  |-  ( ph  ->  C  e.  RR )
4 ioossre 11696 . . . 4  |-  ( C (,) B )  C_  RR
5 dvbdfbdioolem1.d . . . 4  |-  ( ph  ->  D  e.  ( C (,) B ) )
64, 5sseldi 3468 . . 3  |-  ( ph  ->  D  e.  RR )
73rexrd 9689 . . . 4  |-  ( ph  ->  C  e.  RR* )
8 dvbdfbdioolem1.b . . . . 5  |-  ( ph  ->  B  e.  RR )
98rexrd 9689 . . . 4  |-  ( ph  ->  B  e.  RR* )
10 ioogtlb 37180 . . . 4  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  D  e.  ( C (,) B
) )  ->  C  <  D )
117, 9, 5, 10syl3anc 1264 . . 3  |-  ( ph  ->  C  <  D )
12 dvbdfbdioolem1.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
1312rexrd 9689 . . . . 5  |-  ( ph  ->  A  e.  RR* )
14 ioogtlb 37180 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A (,) B
) )  ->  A  <  C )
1513, 9, 2, 14syl3anc 1264 . . . . 5  |-  ( ph  ->  A  <  C )
16 iooltub 37198 . . . . . 6  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  D  e.  ( C (,) B
) )  ->  D  <  B )
177, 9, 5, 16syl3anc 1264 . . . . 5  |-  ( ph  ->  D  <  B )
18 iccssioo 11703 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( A (,) B ) )
1913, 9, 15, 17, 18syl22anc 1265 . . . 4  |-  ( ph  ->  ( C [,] D
)  C_  ( A (,) B ) )
20 dvbdfbdioolem1.f . . . . 5  |-  ( ph  ->  F : ( A (,) B ) --> RR )
21 ax-resscn 9595 . . . . . . 7  |-  RR  C_  CC
2221a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
2320, 22fssd 5755 . . . . . . 7  |-  ( ph  ->  F : ( A (,) B ) --> CC )
241a1i 11 . . . . . . 7  |-  ( ph  ->  ( A (,) B
)  C_  RR )
25 dvbdfbdioolem1.dmdv . . . . . . 7  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
26 dvcn 22752 . . . . . . 7  |-  ( ( ( RR  C_  CC  /\  F : ( A (,) B ) --> CC 
/\  ( A (,) B )  C_  RR )  /\  dom  ( RR 
_D  F )  =  ( A (,) B
) )  ->  F  e.  ( ( A (,) B ) -cn-> CC ) )
2722, 23, 24, 25, 26syl31anc 1267 . . . . . 6  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
28 cncffvrn 21826 . . . . . 6  |-  ( ( RR  C_  CC  /\  F  e.  ( ( A (,) B ) -cn-> CC ) )  ->  ( F  e.  ( ( A (,) B ) -cn-> RR )  <-> 
F : ( A (,) B ) --> RR ) )
2922, 27, 28syl2anc 665 . . . . 5  |-  ( ph  ->  ( F  e.  ( ( A (,) B
) -cn-> RR )  <->  F :
( A (,) B
) --> RR ) )
3020, 29mpbird 235 . . . 4  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> RR ) )
31 rescncf 21825 . . . 4  |-  ( ( C [,] D ) 
C_  ( A (,) B )  ->  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  ( F  |`  ( C [,] D
) )  e.  ( ( C [,] D
) -cn-> RR ) ) )
3219, 30, 31sylc 62 . . 3  |-  ( ph  ->  ( F  |`  ( C [,] D ) )  e.  ( ( C [,] D ) -cn-> RR ) )
3319, 24sstrd 3480 . . . . . . 7  |-  ( ph  ->  ( C [,] D
)  C_  RR )
34 eqid 2429 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3534tgioo2 21732 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3634, 35dvres 22743 . . . . . . 7  |-  ( ( ( RR  C_  CC  /\  F : ( A (,) B ) --> CC )  /\  ( ( A (,) B ) 
C_  RR  /\  ( C [,] D )  C_  RR ) )  ->  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  ( ( RR  _D  F
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( C [,] D ) ) ) )
3722, 23, 24, 33, 36syl22anc 1265 . . . . . 6  |-  ( ph  ->  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) ) ) )
38 iccntr 21750 . . . . . . . 8  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) )  =  ( C (,) D
) )
393, 6, 38syl2anc 665 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) )  =  ( C (,) D
) )
4039reseq2d 5125 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) ) )  =  ( ( RR 
_D  F )  |`  ( C (,) D ) ) )
4137, 40eqtrd 2470 . . . . 5  |-  ( ph  ->  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  ( ( RR 
_D  F )  |`  ( C (,) D ) ) )
4241dmeqd 5057 . . . 4  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  dom  (
( RR  _D  F
)  |`  ( C (,) D ) ) )
4312, 3, 15ltled 9782 . . . . . . 7  |-  ( ph  ->  A  <_  C )
446, 8, 17ltled 9782 . . . . . . 7  |-  ( ph  ->  D  <_  B )
45 ioossioo 11726 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C (,) D )  C_  ( A (,) B ) )
4613, 9, 43, 44, 45syl22anc 1265 . . . . . 6  |-  ( ph  ->  ( C (,) D
)  C_  ( A (,) B ) )
4746, 25sseqtr4d 3507 . . . . 5  |-  ( ph  ->  ( C (,) D
)  C_  dom  ( RR 
_D  F ) )
48 ssdmres 5146 . . . . 5  |-  ( ( C (,) D ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( C (,) D
) )  =  ( C (,) D ) )
4947, 48sylib 199 . . . 4  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( C (,) D ) )  =  ( C (,) D ) )
5042, 49eqtrd 2470 . . 3  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  ( C (,) D ) )
513, 6, 11, 32, 50mvth 22821 . 2  |-  ( ph  ->  E. x  e.  ( C (,) D ) ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `  x )  =  ( ( ( ( F  |`  ( C [,] D ) ) `
 D )  -  ( ( F  |`  ( C [,] D ) ) `  C ) )  /  ( D  -  C ) ) )
5241fveq1d 5883 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `  x )  =  ( ( ( RR  _D  F )  |`  ( C (,) D
) ) `  x
) )
53 fvres 5895 . . . . . . . . 9  |-  ( x  e.  ( C (,) D )  ->  (
( ( RR  _D  F )  |`  ( C (,) D ) ) `
 x )  =  ( ( RR  _D  F ) `  x
) )
5452, 53sylan9eq 2490 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( RR  _D  F ) `
 x ) )
5554eqcomd 2437 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  F ) `  x )  =  ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `
 x ) )
56553adant3 1025 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  (
( RR  _D  F
) `  x )  =  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
) )
57 simp3 1007 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  (
( RR  _D  ( F  |`  ( C [,] D ) ) ) `
 x )  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  (
( F  |`  ( C [,] D ) ) `
 C ) )  /  ( D  -  C ) ) )
586rexrd 9689 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  RR* )
593, 6, 11ltled 9782 . . . . . . . . . . 11  |-  ( ph  ->  C  <_  D )
60 ubicc2 11747 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  D  e.  RR*  /\  C  <_  D )  ->  D  e.  ( C [,] D
) )
617, 58, 59, 60syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  D  e.  ( C [,] D ) )
62 fvres 5895 . . . . . . . . . 10  |-  ( D  e.  ( C [,] D )  ->  (
( F  |`  ( C [,] D ) ) `
 D )  =  ( F `  D
) )
6361, 62syl 17 . . . . . . . . 9  |-  ( ph  ->  ( ( F  |`  ( C [,] D ) ) `  D )  =  ( F `  D ) )
64 lbicc2 11746 . . . . . . . . . . 11  |-  ( ( C  e.  RR*  /\  D  e.  RR*  /\  C  <_  D )  ->  C  e.  ( C [,] D
) )
657, 58, 59, 64syl3anc 1264 . . . . . . . . . 10  |-  ( ph  ->  C  e.  ( C [,] D ) )
66 fvres 5895 . . . . . . . . . 10  |-  ( C  e.  ( C [,] D )  ->  (
( F  |`  ( C [,] D ) ) `
 C )  =  ( F `  C
) )
6765, 66syl 17 . . . . . . . . 9  |-  ( ph  ->  ( ( F  |`  ( C [,] D ) ) `  C )  =  ( F `  C ) )
6863, 67oveq12d 6323 . . . . . . . 8  |-  ( ph  ->  ( ( ( F  |`  ( C [,] D
) ) `  D
)  -  ( ( F  |`  ( C [,] D ) ) `  C ) )  =  ( ( F `  D )  -  ( F `  C )
) )
6968oveq1d 6320 . . . . . . 7  |-  ( ph  ->  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  (
( F  |`  ( C [,] D ) ) `
 C ) )  /  ( D  -  C ) )  =  ( ( ( F `
 D )  -  ( F `  C ) )  /  ( D  -  C ) ) )
70693ad2ant1 1026 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  (
( ( ( F  |`  ( C [,] D
) ) `  D
)  -  ( ( F  |`  ( C [,] D ) ) `  C ) )  / 
( D  -  C
) )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )
7156, 57, 703eqtrd 2474 . . . . 5  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  (
( RR  _D  F
) `  x )  =  ( ( ( F `  D )  -  ( F `  C ) )  / 
( D  -  C
) ) )
72 simp3 1007 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( RR  _D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )
7372eqcomd 2437 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( (
( F `  D
)  -  ( F `
 C ) )  /  ( D  -  C ) )  =  ( ( RR  _D  F ) `  x
) )
7419, 61sseldd 3471 . . . . . . . . . . . . . . 15  |-  ( ph  ->  D  e.  ( A (,) B ) )
7520, 74ffvelrnd 6038 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F `  D
)  e.  RR )
7620, 2ffvelrnd 6038 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F `  C
)  e.  RR )
7775, 76resubcld 10046 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( F `  D )  -  ( F `  C )
)  e.  RR )
7877recnd 9668 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( F `  D )  -  ( F `  C )
)  e.  CC )
79783ad2ant1 1026 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( F `  D )  -  ( F `  C ) )  e.  CC )
80 dvfre 22782 . . . . . . . . . . . . . . . . 17  |-  ( ( F : ( A (,) B ) --> RR 
/\  ( A (,) B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
8120, 24, 80syl2anc 665 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
8225feq2d 5733 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
8381, 82mpbid 213 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
8483adantr 466 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( RR  _D  F ) : ( A (,) B ) --> RR )
8546sselda 3470 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  x  e.  ( A (,) B ) )
8684, 85ffvelrnd 6038 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  F ) `  x )  e.  RR )
8786recnd 9668 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
88873adant3 1025 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
896, 3resubcld 10046 . . . . . . . . . . . . 13  |-  ( ph  ->  ( D  -  C
)  e.  RR )
9089recnd 9668 . . . . . . . . . . . 12  |-  ( ph  ->  ( D  -  C
)  e.  CC )
91903ad2ant1 1026 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( D  -  C )  e.  CC )
923, 6posdifd 10199 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( C  <  D  <->  0  <  ( D  -  C ) ) )
9311, 92mpbid 213 . . . . . . . . . . . . 13  |-  ( ph  ->  0  <  ( D  -  C ) )
9493gt0ne0d 10177 . . . . . . . . . . . 12  |-  ( ph  ->  ( D  -  C
)  =/=  0 )
95943ad2ant1 1026 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( D  -  C )  =/=  0
)
9679, 88, 91, 95divmul3d 10416 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( (
( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) )  =  ( ( RR 
_D  F ) `  x )  <->  ( ( F `  D )  -  ( F `  C ) )  =  ( ( ( RR 
_D  F ) `  x )  x.  ( D  -  C )
) ) )
9773, 96mpbid 213 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( F `  D )  -  ( F `  C ) )  =  ( ( ( RR 
_D  F ) `  x )  x.  ( D  -  C )
) )
9897fveq2d 5885 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( F `  D )  -  ( F `  C )
) )  =  ( abs `  ( ( ( RR  _D  F
) `  x )  x.  ( D  -  C
) ) ) )
9990adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( D  -  C )  e.  CC )
10087, 99absmuld 13494 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( ( ( RR 
_D  F ) `  x )  x.  ( D  -  C )
) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( abs `  ( D  -  C
) ) ) )
1011003adant3 1025 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( ( RR 
_D  F ) `  x )  x.  ( D  -  C )
) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( abs `  ( D  -  C
) ) ) )
10298, 101eqtrd 2470 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( F `  D )  -  ( F `  C )
) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( abs `  ( D  -  C
) ) ) )
1033, 6, 59abssubge0d 13472 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( D  -  C )
)  =  ( D  -  C ) )
104103oveq2d 6321 . . . . . . . 8  |-  ( ph  ->  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( abs `  ( D  -  C
) ) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( D  -  C ) ) )
1051043ad2ant1 1026 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( abs `  ( ( RR 
_D  F ) `  x ) )  x.  ( abs `  ( D  -  C )
) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( D  -  C ) ) )
106102, 105eqtrd 2470 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( F `  D )  -  ( F `  C )
) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( D  -  C ) ) )
10787abscld 13476 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( ( RR  _D  F ) `  x
) )  e.  RR )
108 dvbdfbdioolem1.k . . . . . . . . 9  |-  ( ph  ->  K  e.  RR )
109108adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  K  e.  RR )
11089adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( D  -  C )  e.  RR )
111 0red 9643 . . . . . . . . . 10  |-  ( ph  ->  0  e.  RR )
112111, 89, 93ltled 9782 . . . . . . . . 9  |-  ( ph  ->  0  <_  ( D  -  C ) )
113112adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  0  <_  ( D  -  C ) )
114 dvbdfbdioolem1.dvbd . . . . . . . . . 10  |-  ( ph  ->  A. x  e.  ( A (,) B ) ( abs `  (
( RR  _D  F
) `  x )
)  <_  K )
115114adantr 466 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  A. x  e.  ( A (,) B
) ( abs `  (
( RR  _D  F
) `  x )
)  <_  K )
116 rspa 2799 . . . . . . . . 9  |-  ( ( A. x  e.  ( A (,) B ) ( abs `  (
( RR  _D  F
) `  x )
)  <_  K  /\  x  e.  ( A (,) B ) )  -> 
( abs `  (
( RR  _D  F
) `  x )
)  <_  K )
117115, 85, 116syl2anc 665 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( ( RR  _D  F ) `  x
) )  <_  K
)
118107, 109, 110, 113, 117lemul1ad 10546 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( abs `  ( ( RR 
_D  F ) `  x ) )  x.  ( D  -  C
) )  <_  ( K  x.  ( D  -  C ) ) )
1191183adant3 1025 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( abs `  ( ( RR 
_D  F ) `  x ) )  x.  ( D  -  C
) )  <_  ( K  x.  ( D  -  C ) ) )
120106, 119eqbrtrd 4446 . . . . 5  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( F `  D )  -  ( F `  C )
) )  <_  ( K  x.  ( D  -  C ) ) )
12171, 120syld3an3 1309 . . . 4  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( D  -  C ) ) )
12299abscld 13476 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( D  -  C
) )  e.  RR )
1238, 12resubcld 10046 . . . . . . . . 9  |-  ( ph  ->  ( B  -  A
)  e.  RR )
124123adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( B  -  A )  e.  RR )
12587absge0d 13484 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  0  <_  ( abs `  ( ( RR  _D  F ) `
 x ) ) )
12699absge0d 13484 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  0  <_  ( abs `  ( D  -  C ) ) )
1276, 12, 8, 3, 44, 43le2subd 10232 . . . . . . . . . 10  |-  ( ph  ->  ( D  -  C
)  <_  ( B  -  A ) )
128103, 127eqbrtrd 4446 . . . . . . . . 9  |-  ( ph  ->  ( abs `  ( D  -  C )
)  <_  ( B  -  A ) )
129128adantr 466 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( D  -  C
) )  <_  ( B  -  A )
)
130107, 109, 122, 124, 125, 126, 117, 129lemul12ad 10549 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( abs `  ( ( RR 
_D  F ) `  x ) )  x.  ( abs `  ( D  -  C )
) )  <_  ( K  x.  ( B  -  A ) ) )
1311303adant3 1025 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( abs `  ( ( RR 
_D  F ) `  x ) )  x.  ( abs `  ( D  -  C )
) )  <_  ( K  x.  ( B  -  A ) ) )
132102, 131eqbrtrd 4446 . . . . 5  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( F `  D )  -  ( F `  C )
) )  <_  ( K  x.  ( B  -  A ) ) )
13371, 132syld3an3 1309 . . . 4  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A ) ) )
134121, 133jca 534 . . 3  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  (
( abs `  (
( F `  D
)  -  ( F `
 C ) ) )  <_  ( K  x.  ( D  -  C
) )  /\  ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A ) ) ) )
135134rexlimdv3a 2926 . 2  |-  ( ph  ->  ( E. x  e.  ( C (,) D
) ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
)  ->  ( ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( D  -  C ) )  /\  ( abs `  ( ( F `  D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A
) ) ) ) )
13651, 135mpd 15 1  |-  ( ph  ->  ( ( abs `  (
( F `  D
)  -  ( F `
 C ) ) )  <_  ( K  x.  ( D  -  C
) )  /\  ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   A.wral 2782   E.wrex 2783    C_ wss 3442   class class class wbr 4426   dom cdm 4854   ran crn 4855    |` cres 4856   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538    x. cmul 9543   RR*cxr 9673    < clt 9674    <_ cle 9675    - cmin 9859    / cdiv 10268   (,)cioo 11635   [,]cicc 11638   abscabs 13276   TopOpenctopn 15279   topGenctg 15295  ℂfldccnfld 18905   intcnt 19963   -cn->ccncf 21804    _D cdv 22695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-haus 20262  df-cmp 20333  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699
This theorem is referenced by:  dvbdfbdioolem2  37376  ioodvbdlimc1lem1  37378  ioodvbdlimc1lem2  37379  ioodvbdlimc2lem  37381
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