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Theorem dvbdfbdioolem1 31581
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 11598 . . . 4  |-  ( A (,) B )  C_  RR
2 dvbdfbdioolem1.c . . . 4  |-  ( ph  ->  C  e.  ( A (,) B ) )
31, 2sseldi 3507 . . 3  |-  ( ph  ->  C  e.  RR )
4 ioossre 11598 . . . 4  |-  ( C (,) B )  C_  RR
5 dvbdfbdioolem1.d . . . 4  |-  ( ph  ->  D  e.  ( C (,) B ) )
64, 5sseldi 3507 . . 3  |-  ( ph  ->  D  e.  RR )
73rexrd 9655 . . . 4  |-  ( ph  ->  C  e.  RR* )
8 dvbdfbdioolem1.b . . . . 5  |-  ( ph  ->  B  e.  RR )
98rexrd 9655 . . . 4  |-  ( ph  ->  B  e.  RR* )
10 ioogtlb 31415 . . . 4  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  D  e.  ( C (,) B
) )  ->  C  <  D )
117, 9, 5, 10syl3anc 1228 . . 3  |-  ( ph  ->  C  <  D )
12 dvbdfbdioolem1.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
1312rexrd 9655 . . . . 5  |-  ( ph  ->  A  e.  RR* )
14 ioogtlb 31415 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A (,) B
) )  ->  A  <  C )
1513, 9, 2, 14syl3anc 1228 . . . . 5  |-  ( ph  ->  A  <  C )
16 iooltub 31435 . . . . . 6  |-  ( ( C  e.  RR*  /\  B  e.  RR*  /\  D  e.  ( C (,) B
) )  ->  D  <  B )
177, 9, 5, 16syl3anc 1228 . . . . 5  |-  ( ph  ->  D  <  B )
18 iccssioo 11605 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <  B ) )  ->  ( C [,] D )  C_  ( A (,) B ) )
1913, 9, 15, 17, 18syl22anc 1229 . . . 4  |-  ( ph  ->  ( C [,] D
)  C_  ( A (,) B ) )
20 dvbdfbdioolem1.f . . . . 5  |-  ( ph  ->  F : ( A (,) B ) --> RR )
21 ax-resscn 9561 . . . . . . 7  |-  RR  C_  CC
2221a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
2320, 22fssd 5746 . . . . . . 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 22192 . . . . . . 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 1231 . . . . . 6  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
28 cncffvrn 21270 . . . . . 6  |-  ( ( RR  C_  CC  /\  F  e.  ( ( A (,) B ) -cn-> CC ) )  ->  ( F  e.  ( ( A (,) B ) -cn-> RR )  <-> 
F : ( A (,) B ) --> RR ) )
2922, 27, 28syl2anc 661 . . . . 5  |-  ( ph  ->  ( F  e.  ( ( A (,) B
) -cn-> RR )  <->  F :
( A (,) B
) --> RR ) )
3020, 29mpbird 232 . . . 4  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> RR ) )
31 rescncf 21269 . . . 4  |-  ( ( C [,] D ) 
C_  ( A (,) B )  ->  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  ( F  |`  ( C [,] D
) )  e.  ( ( C [,] D
) -cn-> RR ) ) )
3219, 30, 31sylc 60 . . 3  |-  ( ph  ->  ( F  |`  ( C [,] D ) )  e.  ( ( C [,] D ) -cn-> RR ) )
3319, 24sstrd 3519 . . . . . . 7  |-  ( ph  ->  ( C [,] D
)  C_  RR )
34 eqid 2467 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
3534tgioo2 21176 . . . . . . . 8  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
3634, 35dvres 22183 . . . . . . 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 1229 . . . . . 6  |-  ( ph  ->  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) ) ) )
38 iccntr 21194 . . . . . . . 8  |-  ( ( C  e.  RR  /\  D  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) )  =  ( C (,) D
) )
393, 6, 38syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) )  =  ( C (,) D
) )
4039reseq2d 5279 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( C [,] D ) ) )  =  ( ( RR 
_D  F )  |`  ( C (,) D ) ) )
4137, 40eqtrd 2508 . . . . 5  |-  ( ph  ->  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  ( ( RR 
_D  F )  |`  ( C (,) D ) ) )
4241dmeqd 5211 . . . 4  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  dom  (
( RR  _D  F
)  |`  ( C (,) D ) ) )
4312, 3, 15ltled 9744 . . . . . . 7  |-  ( ph  ->  A  <_  C )
446, 8, 17ltled 9744 . . . . . . 7  |-  ( ph  ->  D  <_  B )
45 ioossioo 11628 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B ) )  ->  ( C (,) D )  C_  ( A (,) B ) )
4613, 9, 43, 44, 45syl22anc 1229 . . . . . 6  |-  ( ph  ->  ( C (,) D
)  C_  ( A (,) B ) )
4725eqcomd 2475 . . . . . 6  |-  ( ph  ->  ( A (,) B
)  =  dom  ( RR  _D  F ) )
4846, 47sseqtrd 3545 . . . . 5  |-  ( ph  ->  ( C (,) D
)  C_  dom  ( RR 
_D  F ) )
49 ssdmres 5301 . . . . 5  |-  ( ( C (,) D ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( C (,) D
) )  =  ( C (,) D ) )
5048, 49sylib 196 . . . 4  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( C (,) D ) )  =  ( C (,) D ) )
51 eqidd 2468 . . . 4  |-  ( ph  ->  ( C (,) D
)  =  ( C (,) D ) )
5242, 50, 513eqtrd 2512 . . 3  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( C [,] D ) ) )  =  ( C (,) D ) )
533, 6, 11, 32, 52mvth 22261 . 2  |-  ( ph  ->  E. x  e.  ( C (,) D ) ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `  x )  =  ( ( ( ( F  |`  ( C [,] D ) ) `
 D )  -  ( ( F  |`  ( C [,] D ) ) `  C ) )  /  ( D  -  C ) ) )
54 nfv 1683 . . 3  |-  F/ x ph
55 nfv 1683 . . . 4  |-  F/ x
( abs `  (
( F `  D
)  -  ( F `
 C ) ) )  <_  ( K  x.  ( D  -  C
) )
56 nfv 1683 . . . 4  |-  F/ x
( abs `  (
( F `  D
)  -  ( F `
 C ) ) )  <_  ( K  x.  ( B  -  A
) )
5755, 56nfan 1875 . . 3  |-  F/ x
( ( abs `  (
( F `  D
)  -  ( F `
 C ) ) )  <_  ( K  x.  ( D  -  C
) )  /\  ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A ) ) )
58 simp1 996 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  ph )
59 simp2 997 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  ( ( F  |`  ( C [,] D
) ) `  C
) )  /  ( D  -  C )
) )  ->  x  e.  ( C (,) D
) )
6041fveq1d 5874 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `  x )  =  ( ( ( RR  _D  F )  |`  ( C (,) D
) ) `  x
) )
6160adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( ( RR  _D  F
)  |`  ( C (,) D ) ) `  x ) )
62 fvres 5886 . . . . . . . . . . 11  |-  ( x  e.  ( C (,) D )  ->  (
( ( RR  _D  F )  |`  ( C (,) D ) ) `
 x )  =  ( ( RR  _D  F ) `  x
) )
6362adantl 466 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( (
( RR  _D  F
)  |`  ( C (,) D ) ) `  x )  =  ( ( RR  _D  F
) `  x )
)
6461, 63eqtrd 2508 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `  x
)  =  ( ( RR  _D  F ) `
 x ) )
6564eqcomd 2475 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  F ) `  x )  =  ( ( RR  _D  ( F  |`  ( C [,] D ) ) ) `
 x ) )
66653adant3 1016 . . . . . . 7  |-  ( (
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
) )
67 simp3 998 . . . . . . 7  |-  ( (
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 ) ) )
686rexrd 9655 . . . . . . . . . . . 12  |-  ( ph  ->  D  e.  RR* )
693, 6, 11ltled 9744 . . . . . . . . . . . 12  |-  ( ph  ->  C  <_  D )
70 ubicc2 11649 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  D  e.  RR*  /\  C  <_  D )  ->  D  e.  ( C [,] D
) )
717, 68, 69, 70syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  D  e.  ( C [,] D ) )
72 fvres 5886 . . . . . . . . . . 11  |-  ( D  e.  ( C [,] D )  ->  (
( F  |`  ( C [,] D ) ) `
 D )  =  ( F `  D
) )
7371, 72syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( F  |`  ( C [,] D ) ) `  D )  =  ( F `  D ) )
74 lbicc2 11648 . . . . . . . . . . . 12  |-  ( ( C  e.  RR*  /\  D  e.  RR*  /\  C  <_  D )  ->  C  e.  ( C [,] D
) )
757, 68, 69, 74syl3anc 1228 . . . . . . . . . . 11  |-  ( ph  ->  C  e.  ( C [,] D ) )
76 fvres 5886 . . . . . . . . . . 11  |-  ( C  e.  ( C [,] D )  ->  (
( F  |`  ( C [,] D ) ) `
 C )  =  ( F `  C
) )
7775, 76syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( F  |`  ( C [,] D ) ) `  C )  =  ( F `  C ) )
7873, 77oveq12d 6313 . . . . . . . . 9  |-  ( ph  ->  ( ( ( F  |`  ( C [,] D
) ) `  D
)  -  ( ( F  |`  ( C [,] D ) ) `  C ) )  =  ( ( F `  D )  -  ( F `  C )
) )
7978oveq1d 6310 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( F  |`  ( C [,] D ) ) `  D )  -  (
( F  |`  ( C [,] D ) ) `
 C ) )  /  ( D  -  C ) )  =  ( ( ( F `
 D )  -  ( F `  C ) )  /  ( D  -  C ) ) )
80793ad2ant1 1017 . . . . . . 7  |-  ( (
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 ) ) )
8166, 67, 803eqtrd 2512 . . . . . 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 )  =  ( ( ( F `  D )  -  ( F `  C ) )  / 
( D  -  C
) ) )
82 simp3 998 . . . . . . . . . . . . . 14  |-  ( (
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 ) ) )
8382eqcomd 2475 . . . . . . . . . . . . 13  |-  ( (
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
) )
8419, 71sseldd 3510 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  D  e.  ( A (,) B ) )
8520, 84ffvelrnd 6033 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F `  D
)  e.  RR )
8620, 2ffvelrnd 6033 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( F `  C
)  e.  RR )
8785, 86resubcld 9999 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( F `  D )  -  ( F `  C )
)  e.  RR )
8887recnd 9634 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( F `  D )  -  ( F `  C )
)  e.  CC )
8988adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( F `  D )  -  ( F `  C ) )  e.  CC )
90893adant3 1016 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( F `  D )  -  ( F `  C ) )  e.  CC )
91 dvfre 22222 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( F : ( A (,) B ) --> RR 
/\  ( A (,) B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
9220, 24, 91syl2anc 661 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
9325feq2d 5724 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
9492, 93mpbid 210 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
9594adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( RR  _D  F ) : ( A (,) B ) --> RR )
9646sselda 3509 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  x  e.  ( A (,) B ) )
9795, 96ffvelrnd 6033 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  F ) `  x )  e.  RR )
9897recnd 9634 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
99983adant3 1016 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
1006, 3resubcld 9999 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( D  -  C
)  e.  RR )
101100recnd 9634 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( D  -  C
)  e.  CC )
102101adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( D  -  C )  e.  CC )
1031023adant3 1016 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( D  -  C )  e.  CC )
104 0red 9609 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  e.  RR )
1053, 6posdifd 10151 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( C  <  D  <->  0  <  ( D  -  C ) ) )
10611, 105mpbid 210 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  0  <  ( D  -  C ) )
107104, 106gtned 9731 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( D  -  C
)  =/=  0 )
108107adantr 465 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( D  -  C )  =/=  0
)
1091083adant3 1016 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( D  -  C )  =/=  0
)
11090, 99, 103, 109divmul3d 10366 . . . . . . . . . . . . 13  |-  ( (
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 )
) ) )
11183, 110mpbid 210 . . . . . . . . . . . 12  |-  ( (
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 )
) )
112111eqcomd 2475 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( (
( RR  _D  F
) `  x )  x.  ( D  -  C
) )  =  ( ( F `  D
)  -  ( F `
 C ) ) )
113112eqcomd 2475 . . . . . . . . . 10  |-  ( (
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 )
) )
114113fveq2d 5876 . . . . . . . . 9  |-  ( (
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
) ) ) )
11598, 102absmuld 13265 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( ( ( RR 
_D  F ) `  x )  x.  ( D  -  C )
) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( abs `  ( D  -  C
) ) ) )
1161153adant3 1016 . . . . . . . . 9  |-  ( (
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
) ) ) )
117114, 116eqtrd 2508 . . . . . . . 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.  ( abs `  ( D  -  C
) ) ) )
1183, 6, 69abssubge0d 13243 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  ( D  -  C )
)  =  ( D  -  C ) )
119118oveq2d 6311 . . . . . . . . 9  |-  ( ph  ->  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( abs `  ( D  -  C
) ) )  =  ( ( abs `  (
( RR  _D  F
) `  x )
)  x.  ( D  -  C ) ) )
1201193ad2ant1 1017 . . . . . . . 8  |-  ( (
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 ) ) )
121117, 120eqtrd 2508 . . . . . . 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.  ( D  -  C ) ) )
12298abscld 13247 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( ( RR  _D  F ) `  x
) )  e.  RR )
123 dvbdfbdioolem1.k . . . . . . . . . 10  |-  ( ph  ->  K  e.  RR )
124123adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  K  e.  RR )
125100adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( D  -  C )  e.  RR )
126104, 100, 106ltled 9744 . . . . . . . . . 10  |-  ( ph  ->  0  <_  ( D  -  C ) )
127126adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  0  <_  ( D  -  C ) )
128 dvbdfbdioolem1.dvbd . . . . . . . . . . 11  |-  ( ph  ->  A. x  e.  ( A (,) B ) ( abs `  (
( RR  _D  F
) `  x )
)  <_  K )
129128adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  A. x  e.  ( A (,) B
) ( abs `  (
( RR  _D  F
) `  x )
)  <_  K )
130 rspa 2834 . . . . . . . . . 10  |-  ( ( A. x  e.  ( A (,) B ) ( abs `  (
( RR  _D  F
) `  x )
)  <_  K  /\  x  e.  ( A (,) B ) )  -> 
( abs `  (
( RR  _D  F
) `  x )
)  <_  K )
131129, 96, 130syl2anc 661 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( ( RR  _D  F ) `  x
) )  <_  K
)
132122, 124, 125, 127, 131lemul1ad 10497 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( abs `  ( ( RR 
_D  F ) `  x ) )  x.  ( D  -  C
) )  <_  ( K  x.  ( D  -  C ) ) )
1331323adant3 1016 . . . . . . 7  |-  ( (
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 ) ) )
134121, 133eqbrtrd 4473 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( F `  D )  -  ( F `  C )
) )  <_  ( K  x.  ( D  -  C ) ) )
13558, 59, 81, 134syl3anc 1228 . . . . 5  |-  ( (
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 ) ) )
136102abscld 13247 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( D  -  C
) )  e.  RR )
1378, 12resubcld 9999 . . . . . . . . . 10  |-  ( ph  ->  ( B  -  A
)  e.  RR )
138137adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( B  -  A )  e.  RR )
13998absge0d 13255 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  0  <_  ( abs `  ( ( RR  _D  F ) `
 x ) ) )
140102absge0d 13255 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  0  <_  ( abs `  ( D  -  C ) ) )
1416, 12, 8, 3, 44, 43le2subd 10183 . . . . . . . . . . 11  |-  ( ph  ->  ( D  -  C
)  <_  ( B  -  A ) )
142118, 141eqbrtrd 4473 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  ( D  -  C )
)  <_  ( B  -  A ) )
143142adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( abs `  ( D  -  C
) )  <_  ( B  -  A )
)
144122, 124, 136, 138, 139, 140, 131, 143lemul12ad 10500 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( C (,) D ) )  ->  ( ( abs `  ( ( RR 
_D  F ) `  x ) )  x.  ( abs `  ( D  -  C )
) )  <_  ( K  x.  ( B  -  A ) ) )
1451443adant3 1016 . . . . . . 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 )
) )  <_  ( K  x.  ( B  -  A ) ) )
146117, 145eqbrtrd 4473 . . . . . 6  |-  ( (
ph  /\  x  e.  ( C (,) D )  /\  ( ( RR 
_D  F ) `  x )  =  ( ( ( F `  D )  -  ( F `  C )
)  /  ( D  -  C ) ) )  ->  ( abs `  ( ( F `  D )  -  ( F `  C )
) )  <_  ( K  x.  ( B  -  A ) ) )
14758, 59, 81, 146syl3anc 1228 . . . . 5  |-  ( (
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 ) ) )
148135, 147jca 532 . . . 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
) )  /\  ( abs `  ( ( F `
 D )  -  ( F `  C ) ) )  <_  ( K  x.  ( B  -  A ) ) ) )
1491483exp 1195 . . 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 ) ) ) ) ) )
15054, 57, 149rexlimd 2951 . 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
) ) ) ) )
15153, 150mpd 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 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    C_ wss 3481   class class class wbr 4453   dom cdm 5005   ran crn 5006    |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509   RR*cxr 9639    < clt 9640    <_ cle 9641    - cmin 9817    / cdiv 10218   (,)cioo 11541   [,]cicc 11544   abscabs 13047   TopOpenctopn 14694   topGenctg 14710  ℂfldccnfld 18290   intcnt 19386   -cn->ccncf 21248    _D cdv 22135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139
This theorem is referenced by:  dvbdfbdioolem2  31582  ioodvbdlimc1lem1  31584  ioodvbdlimc1lem2  31585  ioodvbdlimc2lem  31587  fourierdlem45  31775
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