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Theorem c1liplem1 19833
Description: Lemma for c1lip1 19834. (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 5175 . . . . . 6  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  ran  abs
3 absf 12096 . . . . . . 7  |-  abs : CC
--> RR
4 frn 5556 . . . . . . 7  |-  ( abs
: CC --> RR  ->  ran 
abs  C_  RR )
53, 4ax-mp 8 . . . . . 6  |-  ran  abs  C_  RR
62, 5sstri 3317 . . . . 5  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  RR
76a1i 11 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR )
8 dvf 19747 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
9 ffun 5552 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  Fun  ( RR  _D  F ) )
108, 9ax-mp 8 . . . . . . 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 18876 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( RR 
_D  F )  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 fdm 5554 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) : ( A [,] B
) --> RR  ->  dom  ( ( RR  _D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
1512, 13, 143syl 19 . . . . . . 7  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
16 ssdmres 5127 . . . . . . 7  |-  ( ( A [,] B ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( A [,] B
) )  =  ( A [,] B ) )
1715, 16sylibr 204 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  dom  ( RR 
_D  F ) )
18 c1liplem1.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1918rexrd 9090 . . . . . . 7  |-  ( ph  ->  A  e.  RR* )
20 c1liplem1.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
2120rexrd 9090 . . . . . . 7  |-  ( ph  ->  B  e.  RR* )
22 c1liplem1.le . . . . . . 7  |-  ( ph  ->  A  <_  B )
23 lbicc2 10969 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2419, 21, 22, 23syl3anc 1184 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
25 funfvima2 5933 . . . . . . 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 419 . . . . . 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 1183 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  A
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
28 ffun 5552 . . . . . . 7  |-  ( abs
: CC --> RR  ->  Fun 
abs )
293, 28ax-mp 8 . . . . . 6  |-  Fun  abs
30 imassrn 5175 . . . . . . . 8  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  ran  ( RR  _D  F
)
31 frn 5556 . . . . . . . . 9  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> CC 
->  ran  ( RR  _D  F )  C_  CC )
328, 31ax-mp 8 . . . . . . . 8  |-  ran  ( RR  _D  F )  C_  CC
3330, 32sstri 3317 . . . . . . 7  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  CC
343fdmi 5555 . . . . . . 7  |-  dom  abs  =  CC
3533, 34sseqtr4i 3341 . . . . . 6  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  dom  abs
36 funfvima2 5933 . . . . . 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 654 . . . . 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 3594 . . . . 5  |-  ( ( abs `  ( ( RR  _D  F ) `
 A ) )  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
3927, 37, 383syl 19 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  =/=  (/) )
40 ax-resscn 9003 . . . . . . . 8  |-  RR  C_  CC
41 ssid 3327 . . . . . . . 8  |-  CC  C_  CC
42 cncfss 18882 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4340, 41, 42mp2an 654 . . . . . . 7  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4443, 12sseldi 3306 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
45 cniccbdd 19311 . . . . . 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 1184 . . . . 5  |-  ( ph  ->  E. a  e.  RR  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  x ) )  <_ 
a )
47 fvelima 5737 . . . . . . . . . 10  |-  ( ( Fun  abs  /\  b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
4829, 47mpan 652 . . . . . . . . 9  |-  ( b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
49 fvelima 5737 . . . . . . . . . . . . . 14  |-  ( ( Fun  ( RR  _D  F )  /\  y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
5010, 49mpan 652 . . . . . . . . . . . . 13  |-  ( y  e.  ( ( RR 
_D  F ) "
( A [,] B
) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
51 fvres 5704 . . . . . . . . . . . . . . . . . . 19  |-  ( b  e.  ( A [,] B )  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 b )  =  ( ( RR  _D  F ) `  b
) )
5251adantl 453 . . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x )  =  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )
5554fveq2d 5691 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  b  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  x
) )  =  ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  b ) ) )
5655breq1d 4182 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  b  ->  (
( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  <->  ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )  <_  a )
)
5756rspccva 3011 . . . . . . . . . . . . . . . . 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 4194 . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( abs `  ( ( RR  _D  F ) `  b
) )  =  ( abs `  y ) )
6160breq1d 4182 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( ( abs `  ( ( RR  _D  F ) `
 b ) )  <_  a  <->  ( abs `  y )  <_  a
) )
6259, 61syl5ibcom 212 . . . . . . . . . . . . . 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 2790 . . . . . . . . . . . . 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 30 . . . . . . . . . . . 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 419 . . . . . . . . . . 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 4175 . . . . . . . . . . 11  |-  ( ( abs `  y )  =  b  ->  (
( abs `  y
)  <_  a  <->  b  <_  a ) )
6765, 66syl5ibcom 212 . . . . . . . . . 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 2790 . . . . . . . . 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 30 . . . . . . . 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 2748 . . . . . . 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 424 . . . . . 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 2778 . . . . 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 9924 . . . 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 1184 . . 3  |-  ( ph  ->  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )  e.  RR )
761, 75syl5eqel 2488 . 2  |-  ( ph  ->  K  e.  RR )
77 simplrr 738 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
78 fvres 5704 . . . . . . . . . . 11  |-  ( y  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 y )  =  ( F `  y
) )
7977, 78syl 16 . . . . . . . . . 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 18876 . . . . . . . . . . . . . 14  |-  ( ( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
8280, 81syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( F  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
8382ad2antrr 707 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) ) : ( A [,] B ) --> RR )
8483, 77ffvelrnd 5830 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  RR )
8584recnd 9070 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  CC )
8679, 85eqeltrrd 2479 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  CC )
87 simplrl 737 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
88 fvres 5704 . . . . . . . . . . 11  |-  ( x  e.  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) ) `
 x )  =  ( F `  x
) )
8987, 88syl 16 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  =  ( F `  x ) )
9083, 87ffvelrnd 5830 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  RR )
9190recnd 9070 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  CC )
9289, 91eqeltrrd 2479 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  CC )
9386, 92subcld 9367 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  CC )
94 iccssre 10948 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
9518, 20, 94syl2anc 643 . . . . . . . . . . . 12  |-  ( ph  ->  ( A [,] B
)  C_  RR )
9695ad2antrr 707 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
9796, 77sseldd 3309 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
9896, 87sseldd 3309 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
9997, 98resubcld 9421 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
10099recnd 9070 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  CC )
101 simpr 448 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
102 difrp 10601 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
10398, 97, 102syl2anc 643 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
104101, 103mpbid 202 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR+ )
105104rpne0d 10609 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  =/=  0
)
10693, 100, 105absdivd 12212 . . . . . . 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 707 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
10973ad2antrr 707 . . . . . . . . 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 9746 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  CC )
112111, 34syl6eleqr 2495 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e. 
dom  abs )
11398rexrd 9090 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR* )
11497rexrd 9090 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR* )
11598, 97, 101ltled 9177 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <_  y )
116 ubicc2 10970 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  y  e.  ( x [,] y
) )
117113, 114, 115, 116syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( x [,] y
) )
118 fvres 5704 . . . . . . . . . . . . . 14  |-  ( y  e.  ( x [,] y )  ->  (
( F  |`  (
x [,] y ) ) `  y )  =  ( F `  y ) )
119117, 118syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( x [,] y ) ) `  y )  =  ( F `  y ) )
120 lbicc2 10969 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  x  e.  ( x [,] y
) )
121113, 114, 115, 120syl3anc 1184 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( x [,] y
) )
122 fvres 5704 . . . . . . . . . . . . . 14  |-  ( x  e.  ( x [,] y )  ->  (
( F  |`  (
x [,] y ) ) `  x )  =  ( F `  x ) )
123121, 122syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( x [,] y ) ) `  x )  =  ( F `  x ) )
124119, 123oveq12d 6058 . . . . . . . . . . . 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 6055 . . . . . . . . . . 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 10937 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) )  -> 
( x [,] y
)  C_  ( A [,] B ) )
127126ad2antlr 708 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  ( A [,] B ) )
128 resabs1 5134 . . . . . . . . . . . . . . 15  |-  ( ( x [,] y ) 
C_  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  =  ( F  |`  ( x [,] y ) ) )
129127, 128syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  =  ( F  |`  ( x [,] y ) ) )
13080ad2antrr 707 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> RR ) )
131 rescncf 18880 . . . . . . . . . . . . . . 15  |-  ( ( x [,] y ) 
C_  ( A [,] B )  ->  (
( F  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) ) )
132127, 130, 131sylc 58 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
133129, 132eqeltrrd 2479 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
13440a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  RR  C_  CC )
135 c1liplem1.f . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  F  e.  ( CC 
^pm  RR ) )
136135ad2antrr 707 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F  e.  ( CC  ^pm  RR ) )
137 cnex 9027 . . . . . . . . . . . . . . . . . . . 20  |-  CC  e.  _V
138 reex 9037 . . . . . . . . . . . . . . . . . . . 20  |-  RR  e.  _V
139137, 138elpm2 7004 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
140139simplbi 447 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  F : dom  F --> CC )
141136, 140syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F : dom  F --> CC )
142139simprbi 451 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
143136, 142syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  F  C_  RR )
144 iccssre 10948 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x [,] y
)  C_  RR )
14598, 97, 144syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  RR )
146 eqid 2404 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
147146tgioo2 18787 . . . . . . . . . . . . . . . . . 18  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
148146, 147dvres 19751 . . . . . . . . . . . . . . . . 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 ) ) ) )
149134, 141, 143, 145, 148syl22anc 1185 . . . . . . . . . . . . . . . 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 ) ) ) )
150 iccntr 18805 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
15198, 97, 150syl2anc 643 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (
x [,] y ) )  =  ( x (,) y ) )
152151reseq2d 5105 . . . . . . . . . . . . . . . 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
) ) )
153149, 152eqtrd 2436 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( RR  _D  ( F  |`  (
x [,] y ) ) )  =  ( ( RR  _D  F
)  |`  ( x (,) y ) ) )
154153dmeqd 5031 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( RR 
_D  ( F  |`  ( x [,] y
) ) )  =  dom  ( ( RR 
_D  F )  |`  ( x (,) y
) ) )
155 ioossicc 10952 . . . . . . . . . . . . . . . . 17  |-  ( x (,) y )  C_  ( x [,] y
)
156155, 127syl5ss 3319 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  ( A [,] B ) )
15717ad2antrr 707 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  dom  ( RR  _D  F
) )
158156, 157sstrd 3318 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  dom  ( RR  _D  F
) )
159 ssdmres 5127 . . . . . . . . . . . . . . 15  |-  ( ( x (,) y ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
160158, 159sylib 189 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
161154, 160eqtrd 2436 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( RR 
_D  ( F  |`  ( x [,] y
) ) )  =  ( x (,) y
) )
16298, 97, 101, 133, 161mvth 19829 . . . . . . . . . . . 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
) ) )
163153fveq1d 5689 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( RR  _D  ( F  |`  ( x [,] y
) ) ) `  a )  =  ( ( ( RR  _D  F )  |`  (
x (,) y ) ) `  a ) )
164163adantrr 698 . . . . . . . . . . . . . . . . 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
) )
165 fvres 5704 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  ( x (,) y )  ->  (
( ( RR  _D  F )  |`  (
x (,) y ) ) `  a )  =  ( ( RR 
_D  F ) `  a ) )
166165ad2antll 710 . . . . . . . . . . . . . . . . 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 ) )
167164, 166eqtrd 2436 . . . . . . . . . . . . . . . 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 ) )
16810a1i 11 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  ->  Fun  ( RR  _D  F
) )
16917ad2antrr 707 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( A [,] B
)  C_  dom  ( RR 
_D  F ) )
170156sseld 3307 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( a  e.  ( x (,) y
)  ->  a  e.  ( A [,] B ) ) )
171170impr 603 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
a  e.  ( A [,] B ) )
172 funfvima2 5933 . . . . . . . . . . . . . . . . . 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 ) ) ) )
173172imp 419 . . . . . . . . . . . . . . . . 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 ) ) )
174168, 169, 171, 173syl21anc 1183 . . . . . . . . . . . . . . . 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 ) ) )
175167, 174eqeltrd 2478 . . . . . . . . . . . . . . 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
) ) )
176 eleq1 2464 . . . . . . . . . . . . . . 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
) ) ) )
177175, 176syl5ibcom 212 . . . . . . . . . . . . . 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 ) ) ) )
178177expr 599 . . . . . . . . . . . . 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 ) ) ) ) )
179178rexlimdv 2789 . . . . . . . . . . . 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 ) ) ) )
180162, 179mpd 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 ) ) )
181125, 180eqeltrrd 2479 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )
182 funfvima 5932 . . . . . . . . . . 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
) ) ) ) )
183182imp 419 . . . . . . . . . 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
) ) ) )
184110, 112, 181, 183syl21anc 1183 . . . . . . . . 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
) ) ) )
185 suprub 9925 . . . . . . . . 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 ,  <  ) )
186107, 108, 109, 184, 185syl31anc 1187 . . . . . . . 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 ,  <  ) )
187186, 1syl6breqr 4212 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  <_  K )
188106, 187eqbrtrrd 4194 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( ( F `
 y )  -  ( F `  x ) ) )  /  ( abs `  ( y  -  x ) ) )  <_  K )
18993abscld 12193 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  e.  RR )
19076ad2antrr 707 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  RR )
191100, 105absrpcld 12205 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  RR+ )
192189, 190, 191ledivmuld 10653 . . . . . 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 ) ) )
193188, 192mpbid 202 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  (
( abs `  (
y  -  x ) )  x.  K ) )
194191rpcnd 10606 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  CC )
195190recnd 9070 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  CC )
196194, 195mulcomd 9065 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( y  -  x ) )  x.  K )  =  ( K  x.  ( abs `  ( y  -  x
) ) ) )
197193, 196breqtrd 4196 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  ( K  x.  ( abs `  ( y  -  x
) ) ) )
198197ex 424 . . 3  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  ( K  x.  ( abs `  ( y  -  x ) ) ) ) )
199198ralrimivva 2758 . 2  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( abs `  (
( F `  y
)  -  ( F `
 x ) ) )  <_  ( K  x.  ( abs `  (
y  -  x ) ) ) ) )
20076, 199jca 519 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   A.wral 2666   E.wrex 2667    C_ wss 3280   (/)c0 3588   class class class wbr 4172   dom cdm 4837   ran crn 4838    |` cres 4839   "cima 4840   Fun wfun 5407   -->wf 5409   ` cfv 5413  (class class class)co 6040    ^pm cpm 6978   supcsup 7403   CCcc 8944   RRcr 8945    x. cmul 8951   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247    / cdiv 9633   RR+crp 10568   (,)cioo 10872   [,]cicc 10875   abscabs 11994   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   intcnt 17036   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  c1lip1  19834
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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