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Theorem c1liplem1 21310
Description: Lemma for c1lip1 21311. (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 5168 . . . . . 6  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  ran  abs
3 absf 12809 . . . . . . 7  |-  abs : CC
--> RR
4 frn 5553 . . . . . . 7  |-  ( abs
: CC --> RR  ->  ran 
abs  C_  RR )
53, 4ax-mp 5 . . . . . 6  |-  ran  abs  C_  RR
62, 5sstri 3353 . . . . 5  |-  ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) )  C_  RR
76a1i 11 . . . 4  |-  ( ph  ->  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  C_  RR )
8 dvf 21224 . . . . . . . 8  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
9 ffun 5549 . . . . . . . 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 20311 . . . . . . . 8  |-  ( ( ( RR  _D  F
)  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> RR )  ->  ( ( RR 
_D  F )  |`  ( A [,] B ) ) : ( A [,] B ) --> RR )
14 fdm 5551 . . . . . . . 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 5120 . . . . . . 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 9421 . . . . . . 7  |-  ( ph  ->  A  e.  RR* )
20 c1liplem1.b . . . . . . . 8  |-  ( ph  ->  B  e.  RR )
2120rexrd 9421 . . . . . . 7  |-  ( ph  ->  B  e.  RR* )
22 c1liplem1.le . . . . . . 7  |-  ( ph  ->  A  <_  B )
23 lbicc2 11388 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
2419, 21, 22, 23syl3anc 1211 . . . . . 6  |-  ( ph  ->  A  e.  ( A [,] B ) )
25 funfvima2 5940 . . . . . . 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 429 . . . . . 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 1210 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  A
)  e.  ( ( RR  _D  F )
" ( A [,] B ) ) )
28 ffun 5549 . . . . . . 7  |-  ( abs
: CC --> RR  ->  Fun 
abs )
293, 28ax-mp 5 . . . . . 6  |-  Fun  abs
30 imassrn 5168 . . . . . . . 8  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  ran  ( RR  _D  F
)
31 frn 5553 . . . . . . . . 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 3353 . . . . . . 7  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  CC
343fdmi 5552 . . . . . . 7  |-  dom  abs  =  CC
3533, 34sseqtr4i 3377 . . . . . 6  |-  ( ( RR  _D  F )
" ( A [,] B ) )  C_  dom  abs
36 funfvima2 5940 . . . . . 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 665 . . . . 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 3631 . . . . 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 9327 . . . . . . . 8  |-  RR  C_  CC
41 ssid 3363 . . . . . . . 8  |-  CC  C_  CC
42 cncfss 20317 . . . . . . . 8  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A [,] B
) -cn-> RR )  C_  (
( A [,] B
) -cn-> CC ) )
4340, 41, 42mp2an 665 . . . . . . 7  |-  ( ( A [,] B )
-cn-> RR )  C_  (
( A [,] B
) -cn-> CC )
4443, 12sseldi 3342 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A [,] B ) )  e.  ( ( A [,] B ) -cn-> CC ) )
45 cniccbdd 20787 . . . . . 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 1211 . . . . 5  |-  ( ph  ->  E. a  e.  RR  A. x  e.  ( A [,] B ) ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  x ) )  <_ 
a )
47 fvelima 5731 . . . . . . . . . 10  |-  ( ( Fun  abs  /\  b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
4829, 47mpan 663 . . . . . . . . 9  |-  ( b  e.  ( abs " (
( RR  _D  F
) " ( A [,] B ) ) )  ->  E. y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) ( abs `  y
)  =  b )
49 fvelima 5731 . . . . . . . . . . . . . 14  |-  ( ( Fun  ( RR  _D  F )  /\  y  e.  ( ( RR  _D  F ) " ( A [,] B ) ) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
5010, 49mpan 663 . . . . . . . . . . . . 13  |-  ( y  e.  ( ( RR 
_D  F ) "
( A [,] B
) )  ->  E. b  e.  ( A [,] B
) ( ( RR 
_D  F ) `  b )  =  y )
51 fvres 5692 . . . . . . . . . . . . . . . . . . 19  |-  ( b  e.  ( A [,] B )  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 b )  =  ( ( RR  _D  F ) `  b
) )
5251adantl 463 . . . . . . . . . . . . . . . . . 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 5683 . . . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . . . . . . 20  |-  ( x  =  b  ->  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x )  =  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )
5554fveq2d 5683 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  b  ->  ( abs `  ( ( ( RR  _D  F )  |`  ( A [,] B
) ) `  x
) )  =  ( abs `  ( ( ( RR  _D  F
)  |`  ( A [,] B ) ) `  b ) ) )
5655breq1d 4290 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  b  ->  (
( abs `  (
( ( RR  _D  F )  |`  ( A [,] B ) ) `
 x ) )  <_  a  <->  ( abs `  ( ( ( RR 
_D  F )  |`  ( A [,] B ) ) `  b ) )  <_  a )
)
5756rspccva 3061 . . . . . . . . . . . . . . . . 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 4302 . . . . . . . . . . . . . . . 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 706 . . . . . . . . . . . . . . 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 5679 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  b )  =  y  ->  ( abs `  ( ( RR  _D  F ) `  b
) )  =  ( abs `  y ) )
6160breq1d 4290 . . . . . . . . . . . . . . 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 2831 . . . . . . . . . . . . 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 429 . . . . . . . . . . 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 4283 . . . . . . . . . . 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 2831 . . . . . . . . 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 2788 . . . . . . 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 434 . . . . . 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 2818 . . . . 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 10278 . . . 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 1211 . . 3  |-  ( ph  ->  sup ( ( abs " ( ( RR 
_D  F ) "
( A [,] B
) ) ) ,  RR ,  <  )  e.  RR )
761, 75syl5eqel 2517 . 2  |-  ( ph  ->  K  e.  RR )
77 simplrr 753 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( A [,] B ) )
78 fvres 5692 . . . . . . . . . . 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 20311 . . . . . . . . . . . . . 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 718 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) ) : ( A [,] B ) --> RR )
8483, 77ffvelrnd 5832 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  RR )
8584recnd 9400 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  y )  e.  CC )
8679, 85eqeltrrd 2508 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  y )  e.  CC )
87 simplrl 752 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( A [,] B ) )
88 fvres 5692 . . . . . . . . . . 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 5832 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  RR )
9190recnd 9400 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) ) `  x )  e.  CC )
9289, 91eqeltrrd 2508 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x )  e.  CC )
9386, 92subcld 9707 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F `  y )  -  ( F `  x ) )  e.  CC )
94 iccssre 11365 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
9518, 20, 94syl2anc 654 . . . . . . . . . . . 12  |-  ( ph  ->  ( A [,] B
)  C_  RR )
9695ad2antrr 718 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  RR )
9796, 77sseldd 3345 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR )
9896, 87sseldd 3345 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR )
9997, 98resubcld 9764 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  RR )
10099recnd 9400 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  e.  CC )
101 simpr 458 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <  y )
102 difrp 11012 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x  <  y  <->  ( y  -  x )  e.  RR+ ) )
10398, 97, 102syl2anc 654 . . . . . . . . . 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 11020 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( y  -  x )  =/=  0
)
10693, 100, 105absdivd 12925 . . . . . . 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 718 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs " ( ( RR  _D  F ) " ( A [,] B ) ) )  =/=  (/) )
10973ad2antrr 718 . . . . . . . . 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 10095 . . . . . . . . . . 11  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e.  CC )
112111, 34syl6eleqr 2524 . . . . . . . . . 10  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( (
( F `  y
)  -  ( F `
 x ) )  /  ( y  -  x ) )  e. 
dom  abs )
11398rexrd 9421 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  RR* )
11497rexrd 9421 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  RR* )
11598, 97, 101ltled 9510 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  <_  y )
116 ubicc2 11389 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  y  e.  ( x [,] y
) )
117113, 114, 115, 116syl3anc 1211 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  y  e.  ( x [,] y
) )
118 fvres 5692 . . . . . . . . . . . . . 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 11388 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  RR*  /\  y  e.  RR*  /\  x  <_ 
y )  ->  x  e.  ( x [,] y
) )
121113, 114, 115, 120syl3anc 1211 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  x  e.  ( x [,] y
) )
122 fvres 5692 . . . . . . . . . . . . . 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 6098 . . . . . . . . . . . 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 6095 . . . . . . . . . . 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 11354 . . . . . . . . . . . . . . . 16  |-  ( ( x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) )  -> 
( x [,] y
)  C_  ( A [,] B ) )
127126ad2antlr 719 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  ( A [,] B ) )
128 resabs1 5127 . . . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F  |`  ( A [,] B
) )  e.  ( ( A [,] B
) -cn-> RR ) )
131 rescncf 20315 . . . . . . . . . . . . . . 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 60 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( F  |`  ( A [,] B ) )  |`  ( x [,] y
) )  e.  ( ( x [,] y
) -cn-> RR ) )
133129, 132eqeltrrd 2508 . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  F  e.  ( CC  ^pm  RR ) )
137 cnex 9351 . . . . . . . . . . . . . . . . . . . 20  |-  CC  e.  _V
138 reex 9361 . . . . . . . . . . . . . . . . . . . 20  |-  RR  e.  _V
139137, 138elpm2 7232 . . . . . . . . . . . . . . . . . . 19  |-  ( F  e.  ( CC  ^pm  RR )  <->  ( F : dom  F --> CC  /\  dom  F 
C_  RR ) )
140139simplbi 457 . . . . . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . . . . . . 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 11365 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( x [,] y
)  C_  RR )
14598, 97, 144syl2anc 654 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x [,] y )  C_  RR )
146 eqid 2433 . . . . . . . . . . . . . . . . . 18  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
147146tgioo2 20222 . . . . . . . . . . . . . . . . . 18  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
148146, 147dvres 21228 . . . . . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . . . 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 20240 . . . . . . . . . . . . . . . . . 18  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( x [,] y ) )  =  ( x (,) y
) )
15198, 97, 150syl2anc 654 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  (
x [,] y ) )  =  ( x (,) y ) )
152151reseq2d 5097 . . . . . . . . . . . . . . . 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 2465 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( RR  _D  ( F  |`  (
x [,] y ) ) )  =  ( ( RR  _D  F
)  |`  ( x (,) y ) ) )
154153dmeqd 5029 . . . . . . . . . . . . . 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 11369 . . . . . . . . . . . . . . . . 17  |-  ( x (,) y )  C_  ( x [,] y
)
156155, 127syl5ss 3355 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  ( A [,] B ) )
15717ad2antrr 718 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( A [,] B )  C_  dom  ( RR  _D  F
) )
158156, 157sstrd 3354 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( x (,) y )  C_  dom  ( RR  _D  F
) )
159 ssdmres 5120 . . . . . . . . . . . . . . 15  |-  ( ( x (,) y ) 
C_  dom  ( RR  _D  F )  <->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
160158, 159sylib 196 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  dom  ( ( RR  _D  F )  |`  ( x (,) y
) )  =  ( x (,) y ) )
161154, 160eqtrd 2465 . . . . . . . . . . . . 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 21306 . . . . . . . . . . . 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 5681 . . . . . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . . . . 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 5692 . . . . . . . . . . . . . . . . . 18  |-  ( a  e.  ( x (,) y )  ->  (
( ( RR  _D  F )  |`  (
x (,) y ) ) `  a )  =  ( ( RR 
_D  F ) `  a ) )
166165ad2antll 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 ) `
 a ) )
167164, 166eqtrd 2465 . . . . . . . . . . . . . . . 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 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
( A [,] B
)  C_  dom  ( RR 
_D  F ) )
170156sseld 3343 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( a  e.  ( x (,) y
)  ->  a  e.  ( A [,] B ) ) )
171170impr 614 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  ( x  < 
y  /\  a  e.  ( x (,) y
) ) )  -> 
a  e.  ( A [,] B ) )
172 funfvima2 5940 . . . . . . . . . . . . . . . . . 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 429 . . . . . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . . . 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 2493 . . . . . . . . . . . . . . 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 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 ) ) ) )
178177expr 610 . . . . . . . . . . . . 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 2830 . . . . . . . . . . . 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 2508 . . . . . . . . . 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 5939 . . . . . . . . . . 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 429 . . . . . . . . . 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 1210 . . . . . . . . 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 10279 . . . . . . . . 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 1214 . . . . . . . 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 4320 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( ( F `
 y )  -  ( F `  x ) )  /  ( y  -  x ) ) )  <_  K )
188106, 187eqbrtrrd 4302 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( ( F `
 y )  -  ( F `  x ) ) )  /  ( abs `  ( y  -  x ) ) )  <_  K )
18993abscld 12906 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  e.  RR )
19076ad2antrr 718 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  RR )
191100, 105absrpcld 12918 . . . . . . 7  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  RR+ )
192189, 190, 191ledivmuld 11064 . . . . . 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 210 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  (
( abs `  (
y  -  x ) )  x.  K ) )
194191rpcnd 11017 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( y  -  x
) )  e.  CC )
195190recnd 9400 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  K  e.  CC )
196194, 195mulcomd 9395 . . . . 5  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( ( abs `  ( y  -  x ) )  x.  K )  =  ( K  x.  ( abs `  ( y  -  x
) ) ) )
197193, 196breqtrd 4304 . . . 4  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( abs `  ( ( F `  y )  -  ( F `  x )
) )  <_  ( K  x.  ( abs `  ( y  -  x
) ) ) )
198197ex 434 . . 3  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( abs `  ( ( F `  y )  -  ( F `  x ) ) )  <_  ( K  x.  ( abs `  ( y  -  x ) ) ) ) )
199198ralrimivva 2798 . 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 529 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 369    = wceq 1362    e. wcel 1755    =/= wne 2596   A.wral 2705   E.wrex 2706    C_ wss 3316   (/)c0 3625   class class class wbr 4280   dom cdm 4827   ran crn 4828    |` cres 4829   "cima 4830   Fun wfun 5400   -->wf 5402   ` cfv 5406  (class class class)co 6080    ^pm cpm 7203   supcsup 7678   CCcc 9268   RRcr 9269    x. cmul 9275   RR*cxr 9405    < clt 9406    <_ cle 9407    - cmin 9583    / cdiv 9981   RR+crp 10979   (,)cioo 11288   [,]cicc 11291   abscabs 12707   TopOpenctopn 14343   topGenctg 14359  ℂfldccnfld 17662   intcnt 18463   -cn->ccncf 20294    _D cdv 21180
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349  ax-mulf 9350
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-fi 7649  df-sup 7679  df-oi 7712  df-card 8097  df-cda 8325  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-dec 10744  df-uz 10850  df-q 10942  df-rp 10980  df-xneg 11077  df-xadd 11078  df-xmul 11079  df-ioo 11292  df-ico 11294  df-icc 11295  df-fz 11425  df-fzo 11533  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-starv 14236  df-sca 14237  df-vsca 14238  df-ip 14239  df-tset 14240  df-ple 14241  df-ds 14243  df-unif 14244  df-hom 14245  df-cco 14246  df-rest 14344  df-topn 14345  df-0g 14363  df-gsum 14364  df-topgen 14365  df-pt 14366  df-prds 14369  df-xrs 14423  df-qtop 14428  df-imas 14429  df-xps 14431  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-submnd 15448  df-mulg 15528  df-cntz 15815  df-cmn 16259  df-psmet 17653  df-xmet 17654  df-met 17655  df-bl 17656  df-mopn 17657  df-fbas 17658  df-fg 17659  df-cnfld 17663  df-top 18345  df-bases 18347  df-topon 18348  df-topsp 18349  df-cld 18465  df-ntr 18466  df-cls 18467  df-nei 18544  df-lp 18582  df-perf 18583  df-cn 18673  df-cnp 18674  df-haus 18761  df-cmp 18832  df-tx 18977  df-hmeo 19170  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355  df-xms 19737  df-ms 19738  df-tms 19739  df-cncf 20296  df-limc 21183  df-dv 21184
This theorem is referenced by:  c1lip1  21311
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