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Theorem c1liplem1 22932
Description: Lemma for c1lip1 22933. (Contributed by Stefan O'Rear, 15-Nov-2014.)
Hypotheses
Ref Expression
c1liplem1.a  |-  ( ph  ->  A  e.  RR )
c1liplem1.b  |-  ( ph  ->  B  e.  RR )
c1liplem1.le  |-  ( ph  ->  A  <_  B )
c1liplem1.f  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
c1liplem1.dv  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
c1liplem1.cn  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
c1liplem1.k  |-  K  =  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )
Assertion
Ref Expression
c1liplem1  |-  ( ph  ->  ( K  e.  RR  /\ 
A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( K  x.  ( abs `  (
y  -  x ) ) ) ) ) )
Distinct variable groups:    ph, x, y   
x, A, y    x, B, y    x, F, y
Allowed substitution hints:    K( x, y)

Proof of Theorem c1liplem1
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 c1liplem1.k . . 3  |-  K  =  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )
2 imassrn 5194 . . . . . 6  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  ran  abs
3 absf 13386 . . . . . . 7  |-  abs : CC
--> RR
4 frn 5748 . . . . . . 7  |-  ( abs
: CC --> RR  ->  ran 
abs  C_  RR )
53, 4ax-mp 5 . . . . . 6  |-  ran  abs  C_  RR
62, 5sstri 3473 . . . . 5  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  RR
76a1i 11 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR )
8 dvf 22846 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
9 ffun 5744 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
108, 9ax-mp 5 . . . . . . 7  |-  Fun  ( RR  _D  F )
1110a1i 11 . . . . . 6  |-  ( ph  ->  Fun  ( RR  _D  F ) )
12 c1liplem1.dv . . . . . . . 8  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
13 cncff 21909 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( RR 
_D  F )  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 fdm 5746 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) : ( A [,] B
) --> RR  ->  dom  ( ( RR  _D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
1512, 13, 143syl 18 . . . . . . 7  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
16 ssdmres 5141 . . . . . . 7  |-  ( ( A [,] B ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( A [,] B
) )  =  ( A [,] B ) )
1715, 16sylibr 215 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  dom  ( RR 
_D  F ) )
18 c1liplem1.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1918rexrd 9690 . . . . . . 7  |-  ( ph  ->  A  e.  RR* )
20 c1liplem1.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
2120rexrd 9690 . . . . . . 7  |-  ( ph  ->  B  e.  RR* )
22 c1liplem1.le . . . . . . 7  |-  ( ph  ->  A  <_  B )
23 lbicc2 11748 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2419, 21, 22, 23syl3anc 1264 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
25 funfvima2 6152 . . . . . . 7  |-  ( ( Fun  ( RR  _D  F )  /\  ( A [,] B )  C_  dom  ( RR  _D  F
) )  ->  ( A  e.  ( A [,] B )  ->  (
( RR  _D  F
) `  A )  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ) )
2625imp 430 . . . . . 6  |-  ( ( ( Fun  ( RR 
_D  F )  /\  ( A [,] B ) 
C_  dom  ( RR  _D  F ) )  /\  A  e.  ( A [,] B ) )  -> 
( ( RR  _D  F ) `  A
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
2711, 17, 24, 26syl21anc 1263 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  A
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
28 ffun 5744 . . . . . . 7  |-  ( abs
: CC --> RR  ->  Fun 
abs )
293, 28ax-mp 5 . . . . . 6  |-  Fun  abs
30 imassrn 5194 . . . . . . . 8  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  ran  ( RR  _D  F
)
31 frn 5748 . . . . . . . . 9  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  ran  ( RR  _D  F )  C_  CC )
328, 31ax-mp 5 . . . . . . . 8  |-  ran  ( RR  _D  F )  C_  CC
3330, 32sstri 3473 . . . . . . 7  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  CC
343fdmi 5747 . . . . . . 7  |-  dom  abs  =  CC
3533, 34sseqtr4i 3497 . . . . . 6  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  dom  abs
36 funfvima2 6152 . . . . . 6  |-  ( ( Fun  abs  /\  (
( RR  _D  F
) " ( A [,] B ) ) 
C_  dom  abs )  ->  ( ( ( RR 
_D  F ) `  A )  e.  ( ( RR  _D  F
) " ( A [,] B ) )  ->  ( abs `  (
( RR  _D  F
) `  A )
)  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ) )
3729, 35, 36mp2an 676 . . . . 5  |-  ( ( ( RR  _D  F
) `  A )  e.  ( ( RR  _D  F ) " ( A [,] B ) )  ->  ( abs `  (
( RR  _D  F
) `  A )
)  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
38 ne0i 3767 . . . . 5  |-  ( ( abs `  ( ( RR  _D  F ) `
 A ) )  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
3927, 37, 383syl 18 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  =/=  (/) )
40 ax-resscn 9596 . . . . . . . 8  |-  RR  C_  CC
41 ssid 3483 . . . . . . . 8  |-  CC  C_  CC
42 cncfss 21915 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4340, 41, 42mp2an 676 . . . . . . 7  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4443, 12sseldi 3462 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
45 cniccbdd 22396 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  (
( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )  ->  E. a  e.  RR  A. x  e.  ( A [,] B
) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )
4618, 20, 44, 45syl3anc 1264 . . . . 5  |-  ( ph  ->  E. a  e.  RR  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  x ) )  <_ 
a )
47 fvelima 5929 . . . . . . . . . 10  |-  ( ( Fun  abs  /\  b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
4829, 47mpan 674 . . . . . . . . 9  |-  ( b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
49 fvelima 5929 . . . . . . . . . . . . . 14  |-  ( ( Fun  ( RR  _D  F )  /\  y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
5010, 49mpan 674 . . . . . . . . . . . . 13  |-  ( y  e.  ( ( RR 
_D  F ) "
( A [,] B
) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
51 fvres 5891 . . . . . . . . . . . . . . . . . . 19  |-  ( b  e.  ( A [,] B )  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 b )  =  ( ( RR  _D  F ) `  b
) )
5251adantl 467 . . . . . . . . . . . . . . . . . 18  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 b )  =  ( ( RR  _D  F ) `  b
) )
5352fveq2d 5881 . . . . . . . . . . . . . . . . 17  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  b
) )  =  ( abs `  ( ( RR  _D  F ) `
 b ) ) )
54 fveq2 5877 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x )  =  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )
5554fveq2d 5881 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  b  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  x
) )  =  ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  b ) ) )
5655breq1d 4430 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  b  ->  (
( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  <->  ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )  <_  a )
)
5756rspccva 3181 . . . . . . . . . . . . . . . . 17  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  b
) )  <_  a
)
5853, 57eqbrtrrd 4443 . . . . . . . . . . . . . . . 16  |-  ( ( A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  /\  b  e.  ( A [,] B
) )  ->  ( abs `  ( ( RR 
_D  F ) `  b ) )  <_ 
a )
5958adantll 718 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  b  e.  ( A [,] B ) )  -> 
( abs `  (
( RR  _D  F
) `  b )
)  <_  a )
60 fveq2 5877 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( abs `  ( ( RR  _D  F ) `  b
) )  =  ( abs `  y ) )
6160breq1d 4430 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( ( abs `  ( ( RR  _D  F ) `
 b ) )  <_  a  <->  ( abs `  y )  <_  a
) )
6259, 61syl5ibcom 223 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  b  e.  ( A [,] B ) )  -> 
( ( ( RR 
_D  F ) `  b )  =  y  ->  ( abs `  y
)  <_  a )
)
6362rexlimdva 2917 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y  ->  ( abs `  y
)  <_  a )
)
6450, 63syl5 33 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( y  e.  ( ( RR  _D  F
) " ( A [,] B ) )  ->  ( abs `  y
)  <_  a )
)
6564imp 430 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  y  e.  ( ( RR  _D  F ) "
( A [,] B
) ) )  -> 
( abs `  y
)  <_  a )
66 breq1 4423 . . . . . . . . . . 11  |-  ( ( abs `  y )  =  b  ->  (
( abs `  y
)  <_  a  <->  b  <_  a ) )
6765, 66syl5ibcom 223 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a )  /\  y  e.  ( ( RR  _D  F ) "
( A [,] B
) ) )  -> 
( ( abs `  y
)  =  b  -> 
b  <_  a )
)
6867rexlimdva 2917 . . . . . . . . 9  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b  -> 
b  <_  a )
)
6948, 68syl5 33 . . . . . . . 8  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  ( b  e.  ( abs " ( ( RR  _D  F )
" ( A [,] B ) ) )  ->  b  <_  a
) )
7069ralrimiv 2837 . . . . . . 7  |-  ( ( ( ph  /\  a  e.  RR )  /\  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a )  ->  A. b  e.  ( abs " ( ( RR  _D  F )
" ( A [,] B ) ) ) b  <_  a )
7170ex 435 . . . . . 6  |-  ( (
ph  /\  a  e.  RR )  ->  ( A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  x ) )  <_  a  ->  A. b  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) b  <_  a ) )
7271reximdva 2900 . . . . 5  |-  ( ph  ->  ( E. a  e.  RR  A. x  e.  ( A [,] B
) ( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  ->  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
) )
7346, 72mpd 15 . . . 4  |-  ( ph  ->  E. a  e.  RR  A. b  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) b  <_  a )
74 suprcl 10569 . . . 4  |-  ( ( ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR  /\  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  =/=  (/)  /\  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
)  ->  sup (
( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) ,  RR ,  <  )  e.  RR )
757, 39, 73, 74syl3anc 1264 . . 3  |-  ( ph  ->  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )  e.  RR )
761, 75syl5eqel 2514 . 2  |-  ( ph  ->  K  e.  RR )
77 simplrr 769 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
78 fvres 5891 . . . . . . . . . . 11  |-  ( y  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 y )  =  ( F `  y
) )
7977, 78syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  =  ( F `  y ) )
80 c1liplem1.cn . . . . . . . . . . . . . 14  |-  ( ph  ->  ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR ) )
81 cncff 21909 . . . . . . . . . . . . . 14  |-  ( ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
8280, 81syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
8382ad2antrr 730 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) ) : ( A [,] B ) --> RR )
8483, 77ffvelrnd 6034 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  RR )
8584recnd 9669 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  CC )
8679, 85eqeltrrd 2511 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  CC )
87 simplrl 768 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
88 fvres 5891 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 x )  =  ( F `  x
) )
8987, 88syl 17 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  =  ( F `  x ) )
9083, 87ffvelrnd 6034 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  RR )
9190recnd 9669 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  CC )
9289, 91eqeltrrd 2511 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  CC )
9386, 92subcld 9986 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  CC )
94 iccssre 11716 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
9518, 20, 94syl2anc 665 . . . . . . . . . . . 12  |-  ( ph  ->  ( A [,] B
)  C_  RR )
9695ad2antrr 730 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
9796, 77sseldd 3465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
9896, 87sseldd 3465 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
9997, 98resubcld 10047 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
10099recnd 9669 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  CC )
101 simpr 462 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
102 difrp 11337 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
10398, 97, 102syl2anc 665 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
104101, 103mpbid 213 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR+ )
105104rpne0d 11346 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  =/=  0
)
10693, 100, 105absdivd 13502 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  =  ( ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  /  ( abs `  (
y  -  x ) ) ) )
1076a1i 11 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  C_  RR )
10839ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
10973ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
)
11029a1i 11 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  Fun  abs )
11193, 100, 105divcld 10383 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  CC )
112111, 34syl6eleqr 2521 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e. 
dom  abs )
11398rexrd 9690 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR* )
11497rexrd 9690 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR* )
11598, 97, 101ltled 9783 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <_  y )
116 ubicc2 11749 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  y  e.  ( x [,] y
) )
117113, 114, 115, 116syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( x [,] y
) )
118 fvres 5891 . . . . . . . . . . . . . 14  |-  ( y  e.  ( x [,] y )  ->  (
( F  |`  (
x [,] y ) ) `  y )  =  ( F `  y ) )
119117, 118syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( x [,] y ) ) `  y )  =  ( F `  y ) )
120 lbicc2 11748 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  x  e.  ( x [,] y
) )
121113, 114, 115, 120syl3anc 1264 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( x [,] y
) )
122 fvres 5891 . . . . . . . . . . . . . 14  |-  ( x  e.  ( x [,] y )  ->  (
( F  |`  (
x [,] y ) ) `  x )  =  ( F `  x ) )
123121, 122syl 17 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( x [,] y ) ) `  x )  =  ( F `  x ) )
124119, 123oveq12d 6319 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F  |`  (
x [,] y ) ) `  y )  -  ( ( F  |`  ( x [,] y
) ) `  x
) )  =  ( ( F `  y
)  -  ( F `
 x ) ) )
125124oveq1d 6316 . . . . . . . . . . 11  |-  ( ( ( 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 ) ) )
126 iccss2 11705 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) )  -> 
( x [,] y
)  C_  ( A [,] B ) )
127126ad2antlr 731 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  ( A [,] B ) )
128127resabs1d 5149 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  =  ( F  |`  ( x [,] y ) ) )
12980ad2antrr 730 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> RR ) )
130 rescncf 21913 . . . . . . . . . . . . . . 15  |-  ( ( x [,] y ) 
C_  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) ) )
131127, 129, 130sylc 62 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
132128, 131eqeltrrd 2511 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
13340a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  RR  C_  CC )
134 c1liplem1.f . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
135134ad2antrr 730 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F  e.  ( CC  ^pm  RR ) )
136 cnex 9620 . . . . . . . . . . . . . . . . . . . 20  |-  CC  e.  _V
137 reex 9630 . . . . . . . . . . . . . . . . . . . 20  |-  RR  e.  _V
138136, 137elpm2 7507 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
139138simplbi 461 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  F : dom  F --> CC )
140135, 139syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F : dom  F --> CC )
141138simprbi 465 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
142135, 141syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  F  C_  RR )
143 iccssre 11716 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x [,] y
)  C_  RR )
14498, 97, 143syl2anc 665 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  RR )
145 eqid 2422 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
146145tgioo2 21805 . . . . . . . . . . . . . . . . . 18  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
147145, 146dvres 22850 . . . . . . . . . . . . . . . . 17  |-  ( ( ( RR  C_  CC  /\  F : dom  F --> CC )  /\  ( dom  F  C_  RR  /\  (
x [,] y ) 
C_  RR ) )  ->  ( RR  _D  ( F  |`  ( x [,] y ) ) )  =  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) ) ) )
148133, 140, 142, 144, 147syl22anc 1265 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( RR  _D  ( F  |`  (
x [,] y ) ) )  =  ( ( RR  _D  F
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( x [,] y ) ) ) )
149 iccntr 21823 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
15098, 97, 149syl2anc 665 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (
x [,] y ) )  =  ( x (,) y ) )
151150reseq2d 5120 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) ) )  =  ( ( RR 
_D  F )  |`  ( x (,) y
) ) )
152148, 151eqtrd 2463 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( RR  _D  ( F  |`  (
x [,] y ) ) )  =  ( ( RR  _D  F
)  |`  ( x (,) y ) ) )
153152dmeqd 5052 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( RR 
_D  ( F  |`  ( x [,] y
) ) )  =  dom  ( ( RR 
_D  F )  |`  ( x (,) y
) ) )
154 ioossicc 11720 . . . . . . . . . . . . . . . . 17  |-  ( x (,) y )  C_  ( x [,] y
)
155154, 127syl5ss 3475 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  ( A [,] B ) )
15617ad2antrr 730 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  dom  ( RR  _D  F
) )
157155, 156sstrd 3474 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  dom  ( RR  _D  F
) )
158 ssdmres 5141 . . . . . . . . . . . . . . 15  |-  ( ( x (,) y ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
159157, 158sylib 199 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
160153, 159eqtrd 2463 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( RR 
_D  ( F  |`  ( x [,] y
) ) )  =  ( x (,) y
) )
16198, 97, 101, 132, 160mvth 22928 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  E. a  e.  ( x (,) y
) ( ( RR 
_D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) ) )
162152fveq1d 5879 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( RR  _D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( RR  _D  F )  |`  (
x (,) y ) ) `  a ) )
163162adantrr 721 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `  a )  =  ( ( ( RR  _D  F )  |`  ( x (,) y
) ) `  a
) )
164 fvres 5891 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  ( x (,) y )  ->  (
( ( RR  _D  F )  |`  (
x (,) y ) ) `  a )  =  ( ( RR 
_D  F ) `  a ) )
165164ad2antll 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( ( RR 
_D  F )  |`  ( x (,) y
) ) `  a
)  =  ( ( RR  _D  F ) `
 a ) )
166163, 165eqtrd 2463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `  a )  =  ( ( RR 
_D  F ) `  a ) )
16710a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  ->  Fun  ( RR  _D  F
) )
16817ad2antrr 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( A [,] B
)  C_  dom  ( RR 
_D  F ) )
169155sseld 3463 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( a  e.  ( x (,) y
)  ->  a  e.  ( A [,] B ) ) )
170169impr 623 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
a  e.  ( A [,] B ) )
171 funfvima2 6152 . . . . . . . . . . . . . . . . . 18  |-  ( ( Fun  ( RR  _D  F )  /\  ( A [,] B )  C_  dom  ( RR  _D  F
) )  ->  (
a  e.  ( A [,] B )  -> 
( ( RR  _D  F ) `  a
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) ) )
172171imp 430 . . . . . . . . . . . . . . . . 17  |-  ( ( ( Fun  ( RR 
_D  F )  /\  ( A [,] B ) 
C_  dom  ( RR  _D  F ) )  /\  a  e.  ( A [,] B ) )  -> 
( ( RR  _D  F ) `  a
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
173167, 168, 170, 172syl21anc 1263 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  F ) `  a
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
174166, 173eqeltrd 2510 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `  a )  e.  ( ( RR 
_D  F ) "
( A [,] B
) ) )
175 eleq1 2494 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  ( F  |`  ( x [,] y ) ) ) `
 a )  =  ( ( ( ( F  |`  ( x [,] y ) ) `  y )  -  (
( F  |`  (
x [,] y ) ) `  x ) )  /  ( y  -  x ) )  ->  ( ( ( RR  _D  ( F  |`  ( x [,] y
) ) ) `  a )  e.  ( ( RR  _D  F
) " ( A [,] B ) )  <-> 
( ( ( ( F  |`  ( x [,] y ) ) `  y )  -  (
( F  |`  (
x [,] y ) ) `  x ) )  /  ( y  -  x ) )  e.  ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
176174, 175syl5ibcom 223 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( ( ( RR 
_D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  ->  (
( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) ) ) )
177176expr 618 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( a  e.  ( x (,) y
)  ->  ( (
( RR  _D  ( F  |`  ( x [,] y ) ) ) `
 a )  =  ( ( ( ( F  |`  ( x [,] y ) ) `  y )  -  (
( F  |`  (
x [,] y ) ) `  x ) )  /  ( y  -  x ) )  ->  ( ( ( ( F  |`  (
x [,] y ) ) `  y )  -  ( ( F  |`  ( x [,] y
) ) `  x
) )  /  (
y  -  x ) )  e.  ( ( RR  _D  F )
" ( A [,] B ) ) ) ) )
178177rexlimdv 2915 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( E. a  e.  ( x (,) y ) ( ( RR  _D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  ->  (
( ( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) ) ) )
179161, 178mpd 15 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( ( F  |`  ( x [,] y
) ) `  y
)  -  ( ( F  |`  ( x [,] y ) ) `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) ) )
180125, 179eqeltrrd 2511 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )
181 funfvima 6151 . . . . . . . . . . 11  |-  ( ( Fun  abs  /\  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) )  e.  dom  abs )  ->  ( ( ( ( F `  y )  -  ( F `  x ) )  / 
( y  -  x
) )  e.  ( ( RR  _D  F
) " ( A [,] B ) )  ->  ( abs `  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ) )
182181imp 430 . . . . . . . . . 10  |-  ( ( ( Fun  abs  /\  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) )  e.  dom  abs )  /\  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) )  e.  ( ( RR 
_D  F ) "
( A [,] B
) ) )  -> 
( abs `  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
183110, 112, 180, 182syl21anc 1263 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )
184 suprub 10570 . . . . . . . . 9  |-  ( ( ( ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR  /\  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  =/=  (/)  /\  E. a  e.  RR  A. b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) b  <_  a
)  /\  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  e.  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) )  ->  ( abs `  (
( ( F `  y )  -  ( F `  x )
)  /  ( y  -  x ) ) )  <_  sup (
( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) ,  RR ,  <  ) )
185107, 108, 109, 183, 184syl31anc 1267 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  <_  sup (
( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) ,  RR ,  <  ) )
186185, 1syl6breqr 4461 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  <_  K )
187106, 186eqbrtrrd 4443 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( ( F `
 y )  -  ( F `  x ) ) )  /  ( abs `  ( y  -  x ) ) )  <_  K )
18893abscld 13483 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  e.  RR )
18976ad2antrr 730 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  RR )
190100, 105absrpcld 13495 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  RR+ )
191188, 189, 190ledivmuld 11391 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  /  ( abs `  ( y  -  x
) ) )  <_  K 
<->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( ( abs `  ( y  -  x ) )  x.  K ) ) )
192187, 191mpbid 213 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  (
( abs `  (
y  -  x ) )  x.  K ) )
193190rpcnd 11343 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  CC )
194189recnd 9669 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  CC )
195193, 194mulcomd 9664 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( y  -  x ) )  x.  K )  =  ( K  x.  ( abs `  ( y  -  x
) ) ) )
196192, 195breqtrd 4445 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  ( K  x.  ( abs `  ( y  -  x
) ) ) )
197196ex 435 . . 3  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  ( K  x.  ( abs `  ( y  -  x ) ) ) ) )
198197ralrimivva 2846 . 2  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( K  x.  ( abs `  (
y  -  x ) ) ) ) )
19976, 198jca 534 1  |-  ( ph  ->  ( K  e.  RR  /\ 
A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( K  x.  ( abs `  (
y  -  x ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1868    =/= wne 2618   A.wral 2775   E.wrex 2776    C_ wss 3436   (/)c0 3761   class class class wbr 4420   dom cdm 4849   ran crn 4850    |` cres 4851   "cima 4852   Fun wfun 5591   -->wf 5593   ` cfv 5597  (class class class)co 6301    ^pm cpm 7477   supcsup 7956   CCcc 9537   RRcr 9538    x. cmul 9544   RR*cxr 9674    < clt 9675    <_ cle 9676    - cmin 9860    / cdiv 10269   RR+crp 11302   (,)cioo 11635   [,]cicc 11638   abscabs 13283   TopOpenctopn 15305   topGenctg 15321  ℂfldccnfld 18955   intcnt 20016   -cn->ccncf 21892    _D cdv 22802
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 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619
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 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  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 11785  df-fzo 11916  df-seq 12213  df-exp 12272  df-hash 12515  df-cj 13148  df-re 13149  df-im 13150  df-sqrt 13284  df-abs 13285  df-struct 15108  df-ndx 15109  df-slot 15110  df-base 15111  df-sets 15112  df-ress 15113  df-plusg 15188  df-mulr 15189  df-starv 15190  df-sca 15191  df-vsca 15192  df-ip 15193  df-tset 15194  df-ple 15195  df-ds 15197  df-unif 15198  df-hom 15199  df-cco 15200  df-rest 15306  df-topn 15307  df-0g 15325  df-gsum 15326  df-topgen 15327  df-pt 15328  df-prds 15331  df-xrs 15385  df-qtop 15391  df-imas 15392  df-xps 15395  df-mre 15477  df-mrc 15478  df-acs 15480  df-mgm 16473  df-sgrp 16512  df-mnd 16522  df-submnd 16568  df-mulg 16661  df-cntz 16956  df-cmn 17417  df-psmet 18947  df-xmet 18948  df-met 18949  df-bl 18950  df-mopn 18951  df-fbas 18952  df-fg 18953  df-cnfld 18956  df-top 19905  df-bases 19906  df-topon 19907  df-topsp 19908  df-cld 20018  df-ntr 20019  df-cls 20020  df-nei 20098  df-lp 20136  df-perf 20137  df-cn 20227  df-cnp 20228  df-haus 20315  df-cmp 20386  df-tx 20561  df-hmeo 20754  df-fil 20845  df-fm 20937  df-flim 20938  df-flf 20939  df-xms 21319  df-ms 21320  df-tms 21321  df-cncf 21894  df-limc 22805  df-dv 22806
This theorem is referenced by:  c1lip1  22933
  Copyright terms: Public domain W3C validator