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Theorem dvne0 19887
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 2495 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
x  e.  ran  ( RR  _D  F )  <->  0  e.  ran  ( RR  _D  F
) ) )
32notbid 286 . . . . . . . . . . . 12  |-  ( x  =  0  ->  ( -.  x  e.  ran  ( RR  _D  F
)  <->  -.  0  e.  ran  ( RR  _D  F
) ) )
41, 3syl5ibrcom 214 . . . . . . . . . . 11  |-  ( ph  ->  ( x  =  0  ->  -.  x  e.  ran  ( RR  _D  F
) ) )
54necon2ad 2646 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  x  =/=  0 ) )
65imp 419 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  =/=  0 )
7 dvne0.f . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 18915 . . . . . . . . . . . . . . 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 10984 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1310, 11, 12syl2anc 643 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A [,] B
)  C_  RR )
14 dvfre 19829 . . . . . . . . . . . . . 14  |-  ( ( F : ( A [,] B ) --> RR 
/\  ( A [,] B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
159, 13, 14syl2anc 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
16 frn 5589 . . . . . . . . . . . . 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 3340 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  RR )
19 0re 9083 . . . . . . . . . . 11  |-  0  e.  RR
20 lttri2 9149 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  0  e.  RR )  ->  ( x  =/=  0  <->  ( x  <  0  \/  0  <  x ) ) )
2118, 19, 20sylancl 644 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  =/=  0  <->  (
x  <  0  \/  0  <  x ) ) )
22 0xr 9123 . . . . . . . . . . . . . 14  |-  0  e.  RR*
23 elioomnf 10991 . . . . . . . . . . . . . 14  |-  ( 0  e.  RR*  ->  ( x  e.  (  -oo (,) 0 )  <->  ( x  e.  RR  /\  x  <  0 ) ) )
2422, 23ax-mp 8 . . . . . . . . . . . . 13  |-  ( x  e.  (  -oo (,) 0 )  <->  ( x  e.  RR  /\  x  <  0 ) )
2524baib 872 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  (  -oo (,) 0 )  <->  x  <  0 ) )
26 elrp 10606 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
2726baib 872 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  RR+  <->  0  <  x ) )
2825, 27orbi12d 691 . . . . . . . . . . 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 248 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  =/=  0  <->  (
x  e.  (  -oo (,) 0 )  \/  x  e.  RR+ ) ) )
316, 30mpbid 202 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  e.  (  -oo (,) 0 )  \/  x  e.  RR+ ) )
32 elun 3480 . . . . . . . 8  |-  ( x  e.  ( (  -oo (,) 0 )  u.  RR+ ) 
<->  ( x  e.  ( 
-oo (,) 0 )  \/  x  e.  RR+ )
)
3331, 32sylibr 204 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  ( (  -oo (,) 0 )  u.  RR+ ) )
3433ex 424 . . . . . 6  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  x  e.  ( (  -oo (,) 0 )  u.  RR+ ) ) )
3534ssrdv 3346 . . . . 5  |-  ( ph  ->  ran  ( RR  _D  F )  C_  (
(  -oo (,) 0 )  u.  RR+ ) )
36 disjssun 3677 . . . . 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 212 . . . 4  |-  ( ph  ->  ( ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0
) )  =  (/)  ->  ran  ( RR  _D  F )  C_  RR+ )
)
3837imp 419 . . 3  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  (  -oo (,) 0 ) )  =  (/) )  ->  ran  ( RR  _D  F )  C_  RR+ )
3910adantr 452 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  A  e.  RR )
4011adantr 452 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  B  e.  RR )
417adantr 452 . . . . 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 5573 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
4415, 43mpbid 202 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
45 ffn 5583 . . . . . . . 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 552 . . . . . 6  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( ( RR  _D  F )  Fn  ( A (,) B
)  /\  ran  ( RR 
_D  F )  C_  RR+ ) )
48 df-f 5450 . . . . . 6  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR+  <->  ( ( RR  _D  F )  Fn  ( A (,) B
)  /\  ran  ( RR 
_D  F )  C_  RR+ ) )
4947, 48sylibr 204 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( RR  _D  F ) : ( A (,) B ) -->
RR+ )
5039, 40, 41, 49dvgt0 19880 . . . 4  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F ) )
5150orcd 382 . . 3  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
5238, 51syldan 457 . 2  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  (  -oo (,) 0 ) )  =  (/) )  ->  ( F 
Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
53 n0 3629 . . . 4  |-  ( ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0 ) )  =/=  (/)  <->  E. x  x  e.  ( ran  ( RR 
_D  F )  i^i  (  -oo (,) 0
) ) )
54 elin 3522 . . . . . 6  |-  ( x  e.  ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0
) )  <->  ( x  e.  ran  ( RR  _D  F )  /\  x  e.  (  -oo (,) 0
) ) )
55 fvelrnb 5766 . . . . . . . . 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 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  A  e.  RR )
5811adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  B  e.  RR )
597adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
6046adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  ( RR  _D  F )  Fn  ( A (,) B ) )
6144adantr 452 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  ( RR  _D  F ) : ( A (,) B ) --> RR )
6261ffvelrnda 5862 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  RR )
631ad2antrr 707 . . . . . . . . . . . . . . . . . 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 737 . . . . . . . . . . . . . . . . . . . . . 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 733 . . . . . . . . . . . . . . . . . . . . . 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 10988 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A (,) B )  C_  ( A [,] B )
67 rescncf 18919 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( A (,) B ) 
C_  ( A [,] B )  ->  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( A (,) B
) )  e.  ( ( A (,) B
) -cn-> RR ) ) )
6866, 7, 67mpsyl 61 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> RR ) )
6968ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . 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 9039 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  RR  C_  CC
7170a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  RR  C_  CC )
72 fss 5591 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
739, 70, 72sylancl 644 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  F : ( A [,] B ) --> CC )
7466, 13syl5ss 3351 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( A (,) B
)  C_  RR )
75 eqid 2435 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7675tgioo2 18826 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7775, 76dvres 19790 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1185 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) ) )
79 retop 18787 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( topGen ` 
ran  (,) )  e.  Top
80 iooretop 18792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
81 isopn3i 17138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A (,) B )  e.  ( topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A (,) B ) )  =  ( A (,) B ) )
8279, 80, 81mp2an 654 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A (,) B ) )  =  ( A (,) B )
8382reseq2i 5135 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) )
84 fnresdm 5546 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2479 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8778, 86eqtrd 2467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8887dmeqd 5064 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  dom  ( RR  _D  F ) )
8988, 42eqtrd 2467 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( A (,) B ) )
9089ad2antrr 707 . . . . . . . . . . . . . . . . . . . . . 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 19886 . . . . . . . . . . . . . . . . . . . . 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 707 . . . . . . . . . . . . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . . . . . . . . . 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 5722 . . . . . . . . . . . . . . . . . . . . . 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 6091 . . . . . . . . . . . . . . . . . . . . 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 5089 . . . . . . . . . . . . . . . . . . . . 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 3382 . . . . . . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . . . . . . . . . . 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 10991 . . . . . . . . . . . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( RR  _D  F
) `  y )  e.  (  -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 y )  e.  RR  /\  ( ( RR  _D  F ) `
 y )  <  0 ) )
10299, 101sylib 189 . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . . . . . . . . . . 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 9155 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( RR  _D  F ) `  y
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  y )  <  0  ->  ( ( RR  _D  F ) `  y
)  <_  0 ) )
106104, 19, 105sylancl 644 . . . . . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . . . 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 10967 . . . . . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . . . . . 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 1137 . . . . . . . . . . . . . . . . . . . 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 3341 . . . . . . . . . . . . . . . . . . 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 599 . . . . . . . . . . . . . . . . . 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 170 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  -.  0  <_  ( ( RR  _D  F ) `  z
) )
116 ltnle 9147 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( RR  _D  F ) `  z
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  z )  <  0  <->  -.  0  <_  ( ( RR  _D  F ) `  z ) ) )
11762, 19, 116sylancl 644 . . . . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  <  0
)
119 elioomnf 10991 . . . . . . . . . . . . . . . . 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 8 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  z )  e.  (  -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 z )  e.  RR  /\  ( ( RR  _D  F ) `
 z )  <  0 ) )
12162, 118, 120sylanbrc 646 . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . 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 5886 . . . . . . . . . . . . . 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 646 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  ( RR  _D  F ) : ( A (,) B ) --> (  -oo (,) 0
) )
12557, 58, 59, 124dvlt0 19881 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) )
126125olcd 383 . . . . . . . . . . 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 599 . . . . . . . . . 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 2495 . . . . . . . . . . 11  |-  ( ( ( RR  _D  F
) `  y )  =  x  ->  ( ( ( RR  _D  F
) `  y )  e.  (  -oo (,) 0
)  <->  x  e.  (  -oo (,) 0 ) ) )
129128imbi1d 309 . . . . . . . . . 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 212 . . . . . . . . 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 2822 . . . . . . . 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 207 . . . . . . 7  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  ( x  e.  (  -oo (,) 0
)  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) ) )
133132imp3a 421 . . . . . 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 209 . . . . 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 1646 . . . 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 209 . . 3  |-  ( ph  ->  ( ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0
) )  =/=  (/)  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
137136imp 419 . 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 2676 1  |-  ( ph  ->  ( F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F )  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    u. cun 3310    i^i cin 3311    C_ wss 3312   (/)c0 3620   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   ran crn 4871    |` cres 4872    Fn wfn 5441   -->wf 5442   ` cfv 5446    Isom wiso 5447  (class class class)co 6073   CCcc 8980   RRcr 8981   0cc0 8982    -oocmnf 9110   RR*cxr 9111    < clt 9112    <_ cle 9113   RR+crp 10604   (,)cioo 10908   [,]cicc 10911   TopOpenctopn 13641   topGenctg 13657  ℂfldccnfld 16695   Topctop 16950   intcnt 17073   -cn->ccncf 18898    _D cdv 19742
This theorem is referenced by:  dvne0f1  19888  dvcnvrelem1  19893
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-fi 7408  df-sup 7438  df-oi 7471  df-card 7818  df-cda 8040  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-dec 10375  df-uz 10481  df-q 10567  df-rp 10605  df-xneg 10702  df-xadd 10703  df-xmul 10704  df-ioo 10912  df-ico 10914  df-icc 10915  df-fz 11036  df-fzo 11128  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-starv 13536  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-ds 13543  df-unif 13544  df-hom 13545  df-cco 13546  df-rest 13642  df-topn 13643  df-topgen 13659  df-pt 13660  df-prds 13663  df-xrs 13718  df-0g 13719  df-gsum 13720  df-qtop 13725  df-imas 13726  df-xps 13728  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-submnd 14731  df-mulg 14807  df-cntz 15108  df-cmn 15406  df-psmet 16686  df-xmet 16687  df-met 16688  df-bl 16689  df-mopn 16690  df-fbas 16691  df-fg 16692  df-cnfld 16696  df-top 16955  df-bases 16957  df-topon 16958  df-topsp 16959  df-cld 17075  df-ntr 17076  df-cls 17077  df-nei 17154  df-lp 17192  df-perf 17193  df-cn 17283  df-cnp 17284  df-haus 17371  df-cmp 17442  df-tx 17586  df-hmeo 17779  df-fil 17870  df-fm 17962  df-flim 17963  df-flf 17964  df-xms 18342  df-ms 18343  df-tms 18344  df-cncf 18900  df-limc 19745  df-dv 19746
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