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Theorem dvne0 21609
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 2523 . . . . . . . . . . . . 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 2661 . . . . . . . . . 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 20594 . . . . . . . . . . . . . . 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 11481 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1310, 11, 12syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A [,] B
)  C_  RR )
14 dvfre 21551 . . . . . . . . . . . . . 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 5666 . . . . . . . . . . . . 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 3457 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  RR )
19 0re 9490 . . . . . . . . . . 11  |-  0  e.  RR
20 lttri2 9561 . . . . . . . . . . 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 9534 . . . . . . . . . . . . . 14  |-  0  e.  RR*
23 elioomnf 11494 . . . . . . . . . . . . . 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 896 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  ( -oo (,) 0 )  <->  x  <  0 ) )
26 elrp 11097 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
2726baib 896 . . . . . . . . . . . 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 3598 . . . . . . . 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 3463 . . . . 5  |-  ( ph  ->  ran  ( RR  _D  F )  C_  (
( -oo (,) 0 )  u.  RR+ ) )
36 disjssun 3837 . . . . 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 5648 . . . . . . . . 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 5660 . . . . . . . 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 5523 . . . . . 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 21602 . . . 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 3747 . . . 4  |-  ( ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0 ) )  =/=  (/)  <->  E. x  x  e.  ( ran  ( RR 
_D  F )  i^i  ( -oo (,) 0
) ) )
54 elin 3640 . . . . . 6  |-  ( x  e.  ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0
) )  <->  ( x  e.  ran  ( RR  _D  F )  /\  x  e.  ( -oo (,) 0
) ) )
55 fvelrnb 5841 . . . . . . . . 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 5945 . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . . . 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 755 . . . . . . . . . . . . . . . . . . . . . 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 11485 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A (,) B )  C_  ( A [,] B )
67 rescncf 20598 . . . . . . . . . . . . . . . . . . . . . . . 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 9443 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  RR  C_  CC
7170a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  RR  C_  CC )
72 fss 5668 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
739, 70, 72sylancl 662 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  F : ( A [,] B ) --> CC )
7466, 13syl5ss 3468 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( A (,) B
)  C_  RR )
75 eqid 2451 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7675tgioo2 20505 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7775, 76dvres 21512 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) ) )
79 retop 20465 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( topGen ` 
ran  (,) )  e.  Top
80 iooretop 20470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
81 isopn3i 18811 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5208 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) )
84 fnresdm 5621 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2504 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8778, 86eqtrd 2492 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8887dmeqd 5143 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  dom  ( RR  _D  F ) )
8988, 42eqtrd 2492 . . . . . . . . . . . . . . . . . . . . . . 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 21608 . . . . . . . . . . . . . . . . . . . . 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 5794 . . . . . . . . . . . . . . . . . . . . . 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 5794 . . . . . . . . . . . . . . . . . . . . . 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 6211 . . . . . . . . . . . . . . . . . . . . 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 5168 . . . . . . . . . . . . . . . . . . . . 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 3499 . . . . . . . . . . . . . . . . . . . 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 760 . . . . . . . . . . . . . . . . . . . . . . . 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 11494 . . . . . . . . . . . . . . . . . . . . . . . . 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 9567 . . . . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . . 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 11464 . . . . . . . . . . . . . . . . . . . . . 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 1171 . . . . . . . . . . . . . . . . . . . 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 3458 . . . . . . . . . . . . . . . . . . 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 9558 . . . . . . . . . . . . . . . . . 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 11494 . . . . . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . 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 5971 . . . . . . . . . . . . . 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 21603 . . . . . . . . . . . 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 2523 . . . . . . . . . . 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 2940 . . . . . . . 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 1691 . . . 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 2766 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 965    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796    u. cun 3427    i^i cin 3428    C_ wss 3429   (/)c0 3738   class class class wbr 4393   `'ccnv 4940   dom cdm 4941   ran crn 4942    |` cres 4943    Fn wfn 5514   -->wf 5515   ` cfv 5519    Isom wiso 5520  (class class class)co 6193   CCcc 9384   RRcr 9385   0cc0 9386   -oocmnf 9520   RR*cxr 9521    < clt 9522    <_ cle 9523   RR+crp 11095   (,)cioo 11404   [,]cicc 11407   TopOpenctopn 14471   topGenctg 14487  ℂfldccnfld 17936   Topctop 18623   intcnt 18746   -cn->ccncf 20577    _D cdv 21464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-inf2 7951  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463  ax-pre-sup 9464  ax-addf 9465  ax-mulf 9466
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-iin 4275  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-se 4781  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-isom 5528  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-of 6423  df-om 6580  df-1st 6680  df-2nd 6681  df-supp 6794  df-recs 6935  df-rdg 6969  df-1o 7023  df-2o 7024  df-oadd 7027  df-er 7204  df-map 7319  df-pm 7320  df-ixp 7367  df-en 7414  df-dom 7415  df-sdom 7416  df-fin 7417  df-fsupp 7725  df-fi 7765  df-sup 7795  df-oi 7828  df-card 8213  df-cda 8441  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-div 10098  df-nn 10427  df-2 10484  df-3 10485  df-4 10486  df-5 10487  df-6 10488  df-7 10489  df-8 10490  df-9 10491  df-10 10492  df-n0 10684  df-z 10751  df-dec 10860  df-uz 10966  df-q 11058  df-rp 11096  df-xneg 11193  df-xadd 11194  df-xmul 11195  df-ioo 11408  df-ico 11410  df-icc 11411  df-fz 11548  df-fzo 11659  df-seq 11917  df-exp 11976  df-hash 12214  df-cj 12699  df-re 12700  df-im 12701  df-sqr 12835  df-abs 12836  df-struct 14287  df-ndx 14288  df-slot 14289  df-base 14290  df-sets 14291  df-ress 14292  df-plusg 14362  df-mulr 14363  df-starv 14364  df-sca 14365  df-vsca 14366  df-ip 14367  df-tset 14368  df-ple 14369  df-ds 14371  df-unif 14372  df-hom 14373  df-cco 14374  df-rest 14472  df-topn 14473  df-0g 14491  df-gsum 14492  df-topgen 14493  df-pt 14494  df-prds 14497  df-xrs 14551  df-qtop 14556  df-imas 14557  df-xps 14559  df-mre 14635  df-mrc 14636  df-acs 14638  df-mnd 15526  df-submnd 15576  df-mulg 15659  df-cntz 15946  df-cmn 16392  df-psmet 17927  df-xmet 17928  df-met 17929  df-bl 17930  df-mopn 17931  df-fbas 17932  df-fg 17933  df-cnfld 17937  df-top 18628  df-bases 18630  df-topon 18631  df-topsp 18632  df-cld 18748  df-ntr 18749  df-cls 18750  df-nei 18827  df-lp 18865  df-perf 18866  df-cn 18956  df-cnp 18957  df-haus 19044  df-cmp 19115  df-tx 19260  df-hmeo 19453  df-fil 19544  df-fm 19636  df-flim 19637  df-flf 19638  df-xms 20020  df-ms 20021  df-tms 20022  df-cncf 20579  df-limc 21467  df-dv 21468
This theorem is referenced by:  dvne0f1  21610  dvcnvrelem1  21615
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