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Theorem c1liplem1 22563
Description: Lemma for c1lip1 22564. (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 5336 . . . . . 6  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  ran  abs
3 absf 13252 . . . . . . 7  |-  abs : CC
--> RR
4 frn 5719 . . . . . . 7  |-  ( abs
: CC --> RR  ->  ran 
abs  C_  RR )
53, 4ax-mp 5 . . . . . 6  |-  ran  abs  C_  RR
62, 5sstri 3498 . . . . 5  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  RR
76a1i 11 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR )
8 dvf 22477 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
9 ffun 5715 . . . . . . . 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 21563 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( RR 
_D  F )  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 fdm 5717 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) : ( A [,] B
) --> RR  ->  dom  ( ( RR  _D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
1512, 13, 143syl 20 . . . . . . 7  |-  ( ph  ->  dom  ( ( RR 
_D  F )  |`  ( A [,] B ) )  =  ( A [,] B ) )
16 ssdmres 5283 . . . . . . 7  |-  ( ( A [,] B ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( A [,] B
) )  =  ( A [,] B ) )
1715, 16sylibr 212 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  dom  ( RR 
_D  F ) )
18 c1liplem1.a . . . . . . . 8  |-  ( ph  ->  A  e.  RR )
1918rexrd 9632 . . . . . . 7  |-  ( ph  ->  A  e.  RR* )
20 c1liplem1.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
2120rexrd 9632 . . . . . . 7  |-  ( ph  ->  B  e.  RR* )
22 c1liplem1.le . . . . . . 7  |-  ( ph  ->  A  <_  B )
23 lbicc2 11639 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2419, 21, 22, 23syl3anc 1226 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
25 funfvima2 6123 . . . . . . 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 427 . . . . . 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 1225 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  A
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
28 ffun 5715 . . . . . . 7  |-  ( abs
: CC --> RR  ->  Fun 
abs )
293, 28ax-mp 5 . . . . . 6  |-  Fun  abs
30 imassrn 5336 . . . . . . . 8  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  ran  ( RR  _D  F
)
31 frn 5719 . . . . . . . . 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 3498 . . . . . . 7  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  CC
343fdmi 5718 . . . . . . 7  |-  dom  abs  =  CC
3533, 34sseqtr4i 3522 . . . . . 6  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  dom  abs
36 funfvima2 6123 . . . . . 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 670 . . . . 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 3789 . . . . 5  |-  ( ( abs `  ( ( RR  _D  F ) `
 A ) )  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
3927, 37, 383syl 20 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  =/=  (/) )
40 ax-resscn 9538 . . . . . . . 8  |-  RR  C_  CC
41 ssid 3508 . . . . . . . 8  |-  CC  C_  CC
42 cncfss 21569 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4340, 41, 42mp2an 670 . . . . . . 7  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4443, 12sseldi 3487 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
45 cniccbdd 22039 . . . . . 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 1226 . . . . 5  |-  ( ph  ->  E. a  e.  RR  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  x ) )  <_ 
a )
47 fvelima 5900 . . . . . . . . . 10  |-  ( ( Fun  abs  /\  b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
4829, 47mpan 668 . . . . . . . . 9  |-  ( b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
49 fvelima 5900 . . . . . . . . . . . . . 14  |-  ( ( Fun  ( RR  _D  F )  /\  y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
5010, 49mpan 668 . . . . . . . . . . . . 13  |-  ( y  e.  ( ( RR 
_D  F ) "
( A [,] B
) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
51 fvres 5862 . . . . . . . . . . . . . . . . . . 19  |-  ( b  e.  ( A [,] B )  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 b )  =  ( ( RR  _D  F ) `  b
) )
5251adantl 464 . . . . . . . . . . . . . . . . . 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 5852 . . . . . . . . . . . . . . . . 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 5848 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x )  =  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )
5554fveq2d 5852 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  b  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  x
) )  =  ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  b ) ) )
5655breq1d 4449 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  b  ->  (
( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  <->  ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )  <_  a )
)
5756rspccva 3206 . . . . . . . . . . . . . . . . 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 4461 . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 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 5848 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( abs `  ( ( RR  _D  F ) `  b
) )  =  ( abs `  y ) )
6160breq1d 4449 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( ( abs `  ( ( RR  _D  F ) `
 b ) )  <_  a  <->  ( abs `  y )  <_  a
) )
6259, 61syl5ibcom 220 . . . . . . . . . . . . . 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 2946 . . . . . . . . . . . . 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 32 . . . . . . . . . . . 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 427 . . . . . . . . . . 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 4442 . . . . . . . . . . 11  |-  ( ( abs `  y )  =  b  ->  (
( abs `  y
)  <_  a  <->  b  <_  a ) )
6765, 66syl5ibcom 220 . . . . . . . . . 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 2946 . . . . . . . . 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 32 . . . . . . . 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 2866 . . . . . . 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 432 . . . . . 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 2929 . . . . 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 10498 . . . 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 1226 . . 3  |-  ( ph  ->  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )  e.  RR )
761, 75syl5eqel 2546 . 2  |-  ( ph  ->  K  e.  RR )
77 simplrr 760 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
78 fvres 5862 . . . . . . . . . . 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 21563 . . . . . . . . . . . . . 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 723 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) ) : ( A [,] B ) --> RR )
8483, 77ffvelrnd 6008 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  RR )
8584recnd 9611 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  CC )
8679, 85eqeltrrd 2543 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  CC )
87 simplrl 759 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
88 fvres 5862 . . . . . . . . . . 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 6008 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  RR )
9190recnd 9611 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  CC )
9289, 91eqeltrrd 2543 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  CC )
9386, 92subcld 9922 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  CC )
94 iccssre 11609 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
9518, 20, 94syl2anc 659 . . . . . . . . . . . 12  |-  ( ph  ->  ( A [,] B
)  C_  RR )
9695ad2antrr 723 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
9796, 77sseldd 3490 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
9896, 87sseldd 3490 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
9997, 98resubcld 9983 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
10099recnd 9611 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  CC )
101 simpr 459 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
102 difrp 11255 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
10398, 97, 102syl2anc 659 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
104101, 103mpbid 210 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR+ )
105104rpne0d 11264 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  =/=  0
)
10693, 100, 105absdivd 13368 . . . . . . 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 723 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
10973ad2antrr 723 . . . . . . . . 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 10316 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  CC )
112111, 34syl6eleqr 2553 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e. 
dom  abs )
11398rexrd 9632 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR* )
11497rexrd 9632 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR* )
11598, 97, 101ltled 9722 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <_  y )
116 ubicc2 11640 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  y  e.  ( x [,] y
) )
117113, 114, 115, 116syl3anc 1226 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( x [,] y
) )
118 fvres 5862 . . . . . . . . . . . . . 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 11639 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  x  e.  ( x [,] y
) )
121113, 114, 115, 120syl3anc 1226 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( x [,] y
) )
122 fvres 5862 . . . . . . . . . . . . . 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 6288 . . . . . . . . . . . 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 6285 . . . . . . . . . . 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 11598 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) )  -> 
( x [,] y
)  C_  ( A [,] B ) )
127126ad2antlr 724 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  ( A [,] B ) )
128127resabs1d 5291 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  =  ( F  |`  ( x [,] y ) ) )
12980ad2antrr 723 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> RR ) )
130 rescncf 21567 . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
132128, 131eqeltrrd 2543 . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F  e.  ( CC  ^pm  RR ) )
136 cnex 9562 . . . . . . . . . . . . . . . . . . . 20  |-  CC  e.  _V
137 reex 9572 . . . . . . . . . . . . . . . . . . . 20  |-  RR  e.  _V
138136, 137elpm2 7443 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
139138simplbi 458 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  F : dom  F --> CC )
140135, 139syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F : dom  F --> CC )
141138simprbi 462 . . . . . . . . . . . . . . . . . 18  |-  ( F  e.  ( CC  ^pm  RR )  ->  dom  F  C_  RR )
142135, 141syl 16 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  F  C_  RR )
143 iccssre 11609 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x [,] y
)  C_  RR )
14498, 97, 143syl2anc 659 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  RR )
145 eqid 2454 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
146145tgioo2 21474 . . . . . . . . . . . . . . . . . 18  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
147145, 146dvres 22481 . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . 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 21492 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
15098, 97, 149syl2anc 659 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (
x [,] y ) )  =  ( x (,) y ) )
151150reseq2d 5262 . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( RR  _D  ( F  |`  (
x [,] y ) ) )  =  ( ( RR  _D  F
)  |`  ( x (,) y ) ) )
153152dmeqd 5194 . . . . . . . . . . . . . 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 11613 . . . . . . . . . . . . . . . . 17  |-  ( x (,) y )  C_  ( x [,] y
)
155154, 127syl5ss 3500 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  ( A [,] B ) )
15617ad2antrr 723 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  dom  ( RR  _D  F
) )
157155, 156sstrd 3499 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  dom  ( RR  _D  F
) )
158 ssdmres 5283 . . . . . . . . . . . . . . 15  |-  ( ( x (,) y ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
159157, 158sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
160153, 159eqtrd 2495 . . . . . . . . . . . . 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 22559 . . . . . . . . . . . 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 5850 . . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . . 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 5862 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  ( x (,) y )  ->  (
( ( RR  _D  F )  |`  (
x (,) y ) ) `  a )  =  ( ( RR 
_D  F ) `  a ) )
165164ad2antll 726 . . . . . . . . . . . . . . . . 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 2495 . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( A [,] B
)  C_  dom  ( RR 
_D  F ) )
169155sseld 3488 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( a  e.  ( x (,) y
)  ->  a  e.  ( A [,] B ) ) )
170169impr 617 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
a  e.  ( A [,] B ) )
171 funfvima2 6123 . . . . . . . . . . . . . . . . . 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 427 . . . . . . . . . . . . . . . . 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 1225 . . . . . . . . . . . . . . . 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 2542 . . . . . . . . . . . . . . 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 2526 . . . . . . . . . . . . . . 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 220 . . . . . . . . . . . . . 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 613 . . . . . . . . . . . . 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 2944 . . . . . . . . . . . 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 2543 . . . . . . . . . 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 6122 . . . . . . . . . . 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 427 . . . . . . . . . 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 1225 . . . . . . . . 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 10499 . . . . . . . . 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 1229 . . . . . . . 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 4479 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  <_  K )
187106, 186eqbrtrrd 4461 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( ( F `
 y )  -  ( F `  x ) ) )  /  ( abs `  ( y  -  x ) ) )  <_  K )
18893abscld 13349 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  e.  RR )
18976ad2antrr 723 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  RR )
190100, 105absrpcld 13361 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  RR+ )
191188, 189, 190ledivmuld 11308 . . . . . 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 210 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  (
( abs `  (
y  -  x ) )  x.  K ) )
193190rpcnd 11261 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  CC )
194189recnd 9611 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  CC )
195193, 194mulcomd 9606 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( y  -  x ) )  x.  K )  =  ( K  x.  ( abs `  ( y  -  x
) ) ) )
196192, 195breqtrd 4463 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  ( K  x.  ( abs `  ( y  -  x
) ) ) )
197196ex 432 . . 3  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  ( K  x.  ( abs `  ( y  -  x ) ) ) ) )
198197ralrimivva 2875 . 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 530 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 184    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805    C_ wss 3461   (/)c0 3783   class class class wbr 4439   dom cdm 4988   ran crn 4989    |` cres 4990   "cima 4991   Fun wfun 5564   -->wf 5566   ` cfv 5570  (class class class)co 6270    ^pm cpm 7413   supcsup 7892   CCcc 9479   RRcr 9480    x. cmul 9486   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9796    / cdiv 10202   RR+crp 11221   (,)cioo 11532   [,]cicc 11535   abscabs 13149   TopOpenctopn 14911   topGenctg 14927  ℂfldccnfld 18615   intcnt 19685   -cn->ccncf 21546    _D cdv 22433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-struct 14718  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-starv 14799  df-sca 14800  df-vsca 14801  df-ip 14802  df-tset 14803  df-ple 14804  df-ds 14806  df-unif 14807  df-hom 14808  df-cco 14809  df-rest 14912  df-topn 14913  df-0g 14931  df-gsum 14932  df-topgen 14933  df-pt 14934  df-prds 14937  df-xrs 14991  df-qtop 14996  df-imas 14997  df-xps 14999  df-mre 15075  df-mrc 15076  df-acs 15078  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-submnd 16166  df-mulg 16259  df-cntz 16554  df-cmn 16999  df-psmet 18606  df-xmet 18607  df-met 18608  df-bl 18609  df-mopn 18610  df-fbas 18611  df-fg 18612  df-cnfld 18616  df-top 19566  df-bases 19568  df-topon 19569  df-topsp 19570  df-cld 19687  df-ntr 19688  df-cls 19689  df-nei 19766  df-lp 19804  df-perf 19805  df-cn 19895  df-cnp 19896  df-haus 19983  df-cmp 20054  df-tx 20229  df-hmeo 20422  df-fil 20513  df-fm 20605  df-flim 20606  df-flf 20607  df-xms 20989  df-ms 20990  df-tms 20991  df-cncf 21548  df-limc 22436  df-dv 22437
This theorem is referenced by:  c1lip1  22564
  Copyright terms: Public domain W3C validator