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Theorem dvne0 22537
Description: A function on a closed interval with nonzero derivative is either monotone increasing or monotone decreasing. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
dvne0.a  |-  ( ph  ->  A  e.  RR )
dvne0.b  |-  ( ph  ->  B  e.  RR )
dvne0.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
dvne0.d  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
dvne0.z  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F
) )
Assertion
Ref Expression
dvne0  |-  ( ph  ->  ( F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F )  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )

Proof of Theorem dvne0
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvne0.z . . . . . . . . . . . 12  |-  ( ph  ->  -.  0  e.  ran  ( RR  _D  F
) )
2 eleq1 2529 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
x  e.  ran  ( RR  _D  F )  <->  0  e.  ran  ( RR  _D  F
) ) )
32notbid 294 . . . . . . . . . . . 12  |-  ( x  =  0  ->  ( -.  x  e.  ran  ( RR  _D  F
)  <->  -.  0  e.  ran  ( RR  _D  F
) ) )
41, 3syl5ibrcom 222 . . . . . . . . . . 11  |-  ( ph  ->  ( x  =  0  ->  -.  x  e.  ran  ( RR  _D  F
) ) )
54necon2ad 2670 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  x  =/=  0 ) )
65imp 429 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  =/=  0 )
7 dvne0.f . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 21522 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
97, 8syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : ( A [,] B ) --> RR )
10 dvne0.a . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR )
11 dvne0.b . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  RR )
12 iccssre 11631 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1310, 11, 12syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A [,] B
)  C_  RR )
14 dvfre 22479 . . . . . . . . . . . . . 14  |-  ( ( F : ( A [,] B ) --> RR 
/\  ( A [,] B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
159, 13, 14syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
16 frn 5743 . . . . . . . . . . . . 13  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> RR 
->  ran  ( RR  _D  F )  C_  RR )
1715, 16syl 16 . . . . . . . . . . . 12  |-  ( ph  ->  ran  ( RR  _D  F )  C_  RR )
1817sselda 3499 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  RR )
19 0re 9613 . . . . . . . . . . 11  |-  0  e.  RR
20 lttri2 9684 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  0  e.  RR )  ->  ( x  =/=  0  <->  ( x  <  0  \/  0  <  x ) ) )
2118, 19, 20sylancl 662 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  =/=  0  <->  (
x  <  0  \/  0  <  x ) ) )
22 0xr 9657 . . . . . . . . . . . . . 14  |-  0  e.  RR*
23 elioomnf 11644 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR*  ->  ( x  e.  ( -oo (,) 0 )  <->  ( x  e.  RR  /\  x  <  0 ) ) )
2422, 23ax-mp 5 . . . . . . . . . . . . 13  |-  ( x  e.  ( -oo (,) 0 )  <->  ( x  e.  RR  /\  x  <  0 ) )
2524baib 903 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  ( -oo (,) 0 )  <->  x  <  0 ) )
26 elrp 11247 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
2726baib 903 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  RR+  <->  0  <  x ) )
2825, 27orbi12d 709 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  (
( x  e.  ( -oo (,) 0 )  \/  x  e.  RR+ ) 
<->  ( x  <  0  \/  0  <  x ) ) )
2918, 28syl 16 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
( x  e.  ( -oo (,) 0 )  \/  x  e.  RR+ ) 
<->  ( x  <  0  \/  0  <  x ) ) )
3021, 29bitr4d 256 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  =/=  0  <->  (
x  e.  ( -oo (,) 0 )  \/  x  e.  RR+ ) ) )
316, 30mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  e.  ( -oo (,) 0 )  \/  x  e.  RR+ ) )
32 elun 3641 . . . . . . . 8  |-  ( x  e.  ( ( -oo (,) 0 )  u.  RR+ ) 
<->  ( x  e.  ( -oo (,) 0 )  \/  x  e.  RR+ ) )
3331, 32sylibr 212 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  ( ( -oo (,) 0 )  u.  RR+ ) )
3433ex 434 . . . . . 6  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  x  e.  ( ( -oo (,) 0 )  u.  RR+ ) ) )
3534ssrdv 3505 . . . . 5  |-  ( ph  ->  ran  ( RR  _D  F )  C_  (
( -oo (,) 0 )  u.  RR+ ) )
36 disjssun 3887 . . . . 5  |-  ( ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0 ) )  =  (/)  ->  ( ran  ( RR  _D  F
)  C_  ( ( -oo (,) 0 )  u.  RR+ )  <->  ran  ( RR  _D  F )  C_  RR+ )
)
3735, 36syl5ibcom 220 . . . 4  |-  ( ph  ->  ( ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0
) )  =  (/)  ->  ran  ( RR  _D  F )  C_  RR+ )
)
3837imp 429 . . 3  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  ( -oo (,) 0 ) )  =  (/) )  ->  ran  ( RR  _D  F )  C_  RR+ )
3910adantr 465 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  A  e.  RR )
4011adantr 465 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  B  e.  RR )
417adantr 465 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
42 dvne0.d . . . . . . . . . 10  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
4342feq2d 5724 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
4415, 43mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
45 ffn 5737 . . . . . . . 8  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR  ->  ( RR  _D  F )  Fn  ( A (,) B ) )
4644, 45syl 16 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  Fn  ( A (,) B ) )
4746anim1i 568 . . . . . 6  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( ( RR  _D  F )  Fn  ( A (,) B
)  /\  ran  ( RR 
_D  F )  C_  RR+ ) )
48 df-f 5598 . . . . . 6  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR+  <->  ( ( RR  _D  F )  Fn  ( A (,) B
)  /\  ran  ( RR 
_D  F )  C_  RR+ ) )
4947, 48sylibr 212 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( RR  _D  F ) : ( A (,) B ) -->
RR+ )
5039, 40, 41, 49dvgt0 22530 . . . 4  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F ) )
5150orcd 392 . . 3  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
5238, 51syldan 470 . 2  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  ( -oo (,) 0 ) )  =  (/) )  ->  ( F 
Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
53 n0 3803 . . . 4  |-  ( ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0 ) )  =/=  (/)  <->  E. x  x  e.  ( ran  ( RR 
_D  F )  i^i  ( -oo (,) 0
) ) )
54 elin 3683 . . . . . 6  |-  ( x  e.  ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0
) )  <->  ( x  e.  ran  ( RR  _D  F )  /\  x  e.  ( -oo (,) 0
) ) )
55 fvelrnb 5920 . . . . . . . . 9  |-  ( ( RR  _D  F )  Fn  ( A (,) B )  ->  (
x  e.  ran  ( RR  _D  F )  <->  E. y  e.  ( A (,) B
) ( ( RR 
_D  F ) `  y )  =  x ) )
5646, 55syl 16 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  <->  E. y  e.  ( A (,) B ) ( ( RR  _D  F ) `  y
)  =  x ) )
5710adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  A  e.  RR )
5811adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  B  e.  RR )
597adantr 465 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
6046adantr 465 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  ( RR  _D  F )  Fn  ( A (,) B
) )
6144adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  ( RR  _D  F ) : ( A (,) B
) --> RR )
6261ffvelrnda 6032 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  RR )
631ad2antrr 725 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  -.  0  e.  ran  ( RR  _D  F ) )
64 simplrl 761 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  y  e.  ( A (,) B
) )
65 simprl 756 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  z  e.  ( A (,) B
) )
66 ioossicc 11635 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A (,) B )  C_  ( A [,] B )
67 rescncf 21526 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( A (,) B ) 
C_  ( A [,] B )  ->  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( A (,) B
) )  e.  ( ( A (,) B
) -cn-> RR ) ) )
6866, 7, 67mpsyl 63 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> RR ) )
6968ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> RR ) )
70 ax-resscn 9566 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  RR  C_  CC
7170a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  RR  C_  CC )
72 fss 5745 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
739, 70, 72sylancl 662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  F : ( A [,] B ) --> CC )
7466, 13syl5ss 3510 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( A (,) B
)  C_  RR )
75 eqid 2457 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7675tgioo2 21433 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7775, 76dvres 22440 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( RR  C_  CC  /\  F : ( A [,] B ) --> CC )  /\  ( ( A [,] B ) 
C_  RR  /\  ( A (,) B )  C_  RR ) )  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( ( RR  _D  F
)  |`  ( ( int `  ( topGen `  ran  (,) )
) `  ( A (,) B ) ) ) )
7871, 73, 13, 74, 77syl22anc 1229 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) ) )
79 retop 21393 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( topGen ` 
ran  (,) )  e.  Top
80 iooretop 21398 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
81 isopn3i 19709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A (,) B )  e.  ( topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A (,) B ) )  =  ( A (,) B ) )
8279, 80, 81mp2an 672 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A (,) B ) )  =  ( A (,) B )
8382reseq2i 5280 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) )
84 fnresdm 5696 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( RR  _D  F )  Fn  ( A (,) B )  ->  (
( RR  _D  F
)  |`  ( A (,) B ) )  =  ( RR  _D  F
) )
8546, 84syl 16 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( RR  _D  F ) )
8683, 85syl5eq 2510 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8778, 86eqtrd 2498 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8887dmeqd 5215 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  dom  ( RR  _D  F ) )
8988, 42eqtrd 2498 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( A (,) B ) )
9089ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( A (,) B ) )
9164, 65, 69, 90dvivth 22536 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( ( RR  _D  ( F  |`  ( A (,) B ) ) ) `  y ) [,] ( ( RR 
_D  ( F  |`  ( A (,) B ) ) ) `  z
) )  C_  ran  ( RR  _D  ( F  |`  ( A (,) B ) ) ) )
9287ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( RR  _D  F ) )
9392fveq1d 5874 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( RR  _D  ( F  |`  ( A (,) B ) ) ) `
 y )  =  ( ( RR  _D  F ) `  y
) )
9492fveq1d 5874 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( RR  _D  ( F  |`  ( A (,) B ) ) ) `
 z )  =  ( ( RR  _D  F ) `  z
) )
9593, 94oveq12d 6314 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( ( RR  _D  ( F  |`  ( A (,) B ) ) ) `  y ) [,] ( ( RR 
_D  ( F  |`  ( A (,) B ) ) ) `  z
) )  =  ( ( ( RR  _D  F ) `  y
) [,] ( ( RR  _D  F ) `
 z ) ) )
9692rneqd 5240 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  ran  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ran  ( RR 
_D  F ) )
9791, 95, 963sstr3d 3541 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( ( RR  _D  F ) `  y
) [,] ( ( RR  _D  F ) `
 z ) ) 
C_  ran  ( RR  _D  F ) )
9819a1i 11 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  0  e.  RR )
99 simplrr 762 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( RR  _D  F
) `  y )  e.  ( -oo (,) 0
) )
100 elioomnf 11644 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 0  e.  RR*  ->  ( ( ( RR  _D  F
) `  y )  e.  ( -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 y )  e.  RR  /\  ( ( RR  _D  F ) `
 y )  <  0 ) ) )
10122, 100ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( RR  _D  F
) `  y )  e.  ( -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 y )  e.  RR  /\  ( ( RR  _D  F ) `
 y )  <  0 ) )
10299, 101sylib 196 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( ( RR  _D  F ) `  y
)  e.  RR  /\  ( ( RR  _D  F ) `  y
)  <  0 ) )
103102simprd 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( RR  _D  F
) `  y )  <  0 )
104102simpld 459 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( RR  _D  F
) `  y )  e.  RR )
105 ltle 9690 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( RR  _D  F ) `  y
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  y )  <  0  ->  ( ( RR  _D  F ) `  y
)  <_  0 ) )
106104, 19, 105sylancl 662 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( ( RR  _D  F ) `  y
)  <  0  ->  ( ( RR  _D  F
) `  y )  <_  0 ) )
107103, 106mpd 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( RR  _D  F
) `  y )  <_  0 )
108 simprr 757 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  0  <_  ( ( RR  _D  F ) `  z
) )
10965, 62syldan 470 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
( RR  _D  F
) `  z )  e.  RR )
110 elicc2 11614 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( RR  _D  F ) `  y
)  e.  RR  /\  ( ( RR  _D  F ) `  z
)  e.  RR )  ->  ( 0  e.  ( ( ( RR 
_D  F ) `  y ) [,] (
( RR  _D  F
) `  z )
)  <->  ( 0  e.  RR  /\  ( ( RR  _D  F ) `
 y )  <_ 
0  /\  0  <_  ( ( RR  _D  F
) `  z )
) ) )
111104, 109, 110syl2anc 661 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  (
0  e.  ( ( ( RR  _D  F
) `  y ) [,] ( ( RR  _D  F ) `  z
) )  <->  ( 0  e.  RR  /\  (
( RR  _D  F
) `  y )  <_  0  /\  0  <_ 
( ( RR  _D  F ) `  z
) ) ) )
11298, 107, 108, 111mpbir3and 1179 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  0  e.  ( ( ( RR 
_D  F ) `  y ) [,] (
( RR  _D  F
) `  z )
) )
11397, 112sseldd 3500 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  ( z  e.  ( A (,) B
)  /\  0  <_  ( ( RR  _D  F
) `  z )
) )  ->  0  e.  ran  ( RR  _D  F ) )
114113expr 615 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( 0  <_  ( ( RR 
_D  F ) `  z )  ->  0  e.  ran  ( RR  _D  F ) ) )
11563, 114mtod 177 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  -.  0  <_  ( ( RR  _D  F ) `  z
) )
116 ltnle 9681 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( RR  _D  F ) `  z
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  z )  <  0  <->  -.  0  <_  ( ( RR  _D  F ) `  z ) ) )
11762, 19, 116sylancl 662 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  z )  <  0  <->  -.  0  <_  ( ( RR  _D  F
) `  z )
) )
118115, 117mpbird 232 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  <  0
)
119 elioomnf 11644 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  RR*  ->  ( ( ( RR  _D  F
) `  z )  e.  ( -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 z )  e.  RR  /\  ( ( RR  _D  F ) `
 z )  <  0 ) ) )
12022, 119ax-mp 5 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  z )  e.  ( -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 z )  e.  RR  /\  ( ( RR  _D  F ) `
 z )  <  0 ) )
12162, 118, 120sylanbrc 664 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  ( -oo (,) 0 ) )
122121ralrimiva 2871 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  A. z  e.  ( A (,) B
) ( ( RR 
_D  F ) `  z )  e.  ( -oo (,) 0 ) )
123 ffnfv 6058 . . . . . . . . . . . . . 14  |-  ( ( RR  _D  F ) : ( A (,) B ) --> ( -oo (,) 0 )  <->  ( ( RR  _D  F )  Fn  ( A (,) B
)  /\  A. z  e.  ( A (,) B
) ( ( RR 
_D  F ) `  z )  e.  ( -oo (,) 0 ) ) )
12460, 122, 123sylanbrc 664 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  ( RR  _D  F ) : ( A (,) B
) --> ( -oo (,) 0 ) )
12557, 58, 59, 124dvlt0 22531 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  F  Isom  <  ,  `'  <  ( ( A [,] B
) ,  ran  F
) )
126125olcd 393 . . . . . . . . . . 11  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( -oo (,) 0 ) ) )  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
127126expr 615 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  y )  e.  ( -oo (,) 0
)  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
128 eleq1 2529 . . . . . . . . . . 11  |-  ( ( ( RR  _D  F
) `  y )  =  x  ->  ( ( ( RR  _D  F
) `  y )  e.  ( -oo (,) 0
)  <->  x  e.  ( -oo (,) 0 ) ) )
129128imbi1d 317 . . . . . . . . . 10  |-  ( ( ( RR  _D  F
) `  y )  =  x  ->  ( ( ( ( RR  _D  F ) `  y
)  e.  ( -oo (,) 0 )  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )  <->  ( x  e.  ( -oo (,) 0
)  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) ) )
130127, 129syl5ibcom 220 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( A (,) B ) )  ->  ( (
( RR  _D  F
) `  y )  =  x  ->  ( x  e.  ( -oo (,) 0 )  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) ) )
131130rexlimdva 2949 . . . . . . . 8  |-  ( ph  ->  ( E. y  e.  ( A (,) B
) ( ( RR 
_D  F ) `  y )  =  x  ->  ( x  e.  ( -oo (,) 0
)  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) ) )
13256, 131sylbid 215 . . . . . . 7  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  ( x  e.  ( -oo (,) 0
)  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) ) )
133132impd 431 . . . . . 6  |-  ( ph  ->  ( ( x  e. 
ran  ( RR  _D  F )  /\  x  e.  ( -oo (,) 0
) )  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
13454, 133syl5bi 217 . . . . 5  |-  ( ph  ->  ( x  e.  ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0 ) )  ->  ( F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F )  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
135134exlimdv 1725 . . . 4  |-  ( ph  ->  ( E. x  x  e.  ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0
) )  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
13653, 135syl5bi 217 . . 3  |-  ( ph  ->  ( ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0
) )  =/=  (/)  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
137136imp 429 . 2  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  ( -oo (,) 0 ) )  =/=  (/) )  ->  ( F 
Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
13852, 137pm2.61dane 2775 1  |-  ( ph  ->  ( F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F )  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395   E.wex 1613    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    u. cun 3469    i^i cin 3470    C_ wss 3471   (/)c0 3793   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   ran crn 5009    |` cres 5010    Fn wfn 5589   -->wf 5590   ` cfv 5594    Isom wiso 5595  (class class class)co 6296   CCcc 9507   RRcr 9508   0cc0 9509   -oocmnf 9643   RR*cxr 9644    < clt 9645    <_ cle 9646   RR+crp 11245   (,)cioo 11554   [,]cicc 11557   TopOpenctopn 14838   topGenctg 14854  ℂfldccnfld 18546   Topctop 19520   intcnt 19644   -cn->ccncf 21505    _D cdv 22392
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588  ax-mulf 9589
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-iin 4335  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-supp 6918  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-ixp 7489  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-fsupp 7848  df-fi 7889  df-sup 7919  df-oi 7953  df-card 8337  df-cda 8565  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-uz 11107  df-q 11208  df-rp 11246  df-xneg 11343  df-xadd 11344  df-xmul 11345  df-ioo 11558  df-ico 11560  df-icc 11561  df-fz 11698  df-fzo 11821  df-seq 12110  df-exp 12169  df-hash 12408  df-cj 12943  df-re 12944  df-im 12945  df-sqrt 13079  df-abs 13080  df-struct 14645  df-ndx 14646  df-slot 14647  df-base 14648  df-sets 14649  df-ress 14650  df-plusg 14724  df-mulr 14725  df-starv 14726  df-sca 14727  df-vsca 14728  df-ip 14729  df-tset 14730  df-ple 14731  df-ds 14733  df-unif 14734  df-hom 14735  df-cco 14736  df-rest 14839  df-topn 14840  df-0g 14858  df-gsum 14859  df-topgen 14860  df-pt 14861  df-prds 14864  df-xrs 14918  df-qtop 14923  df-imas 14924  df-xps 14926  df-mre 15002  df-mrc 15003  df-acs 15005  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-submnd 16093  df-mulg 16186  df-cntz 16481  df-cmn 16926  df-psmet 18537  df-xmet 18538  df-met 18539  df-bl 18540  df-mopn 18541  df-fbas 18542  df-fg 18543  df-cnfld 18547  df-top 19525  df-bases 19527  df-topon 19528  df-topsp 19529  df-cld 19646  df-ntr 19647  df-cls 19648  df-nei 19725  df-lp 19763  df-perf 19764  df-cn 19854  df-cnp 19855  df-haus 19942  df-cmp 20013  df-tx 20188  df-hmeo 20381  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566  df-xms 20948  df-ms 20949  df-tms 20950  df-cncf 21507  df-limc 22395  df-dv 22396
This theorem is referenced by:  dvne0f1  22538  dvcnvrelem1  22543
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