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Theorem dvne0 19374
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 2356 . . . . . . . . . . . . 13  |-  ( x  =  0  ->  (
x  e.  ran  ( RR  _D  F )  <->  0  e.  ran  ( RR  _D  F
) ) )
32notbid 285 . . . . . . . . . . . 12  |-  ( x  =  0  ->  ( -.  x  e.  ran  ( RR  _D  F
)  <->  -.  0  e.  ran  ( RR  _D  F
) ) )
41, 3syl5ibrcom 213 . . . . . . . . . . 11  |-  ( ph  ->  ( x  =  0  ->  -.  x  e.  ran  ( RR  _D  F
) ) )
54necon2ad 2507 . . . . . . . . . 10  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  x  =/=  0 ) )
65imp 418 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  =/=  0 )
7 dvne0.f . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 18413 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
97, 8syl 15 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : ( A [,] B ) --> RR )
10 dvne0.a . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  RR )
11 dvne0.b . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  RR )
12 iccssre 10747 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1310, 11, 12syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A [,] B
)  C_  RR )
14 dvfre 19316 . . . . . . . . . . . . . 14  |-  ( ( F : ( A [,] B ) --> RR 
/\  ( A [,] B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
159, 13, 14syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
16 frn 5411 . . . . . . . . . . . . 13  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> RR 
->  ran  ( RR  _D  F )  C_  RR )
1715, 16syl 15 . . . . . . . . . . . 12  |-  ( ph  ->  ran  ( RR  _D  F )  C_  RR )
1817sselda 3193 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  RR )
19 0re 8854 . . . . . . . . . . 11  |-  0  e.  RR
20 lttri2 8920 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  0  e.  RR )  ->  ( x  =/=  0  <->  ( x  <  0  \/  0  <  x ) ) )
2118, 19, 20sylancl 643 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  =/=  0  <->  (
x  <  0  \/  0  <  x ) ) )
22 0xr 8894 . . . . . . . . . . . . . 14  |-  0  e.  RR*
23 elioomnf 10754 . . . . . . . . . . . . . 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 871 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  (  -oo (,) 0 )  <->  x  <  0 ) )
26 elrp 10372 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
2726baib 871 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  RR+  <->  0  <  x ) )
2825, 27orbi12d 690 . . . . . . . . . . 11  |-  ( x  e.  RR  ->  (
( x  e.  ( 
-oo (,) 0 )  \/  x  e.  RR+ )  <->  ( x  <  0  \/  0  <  x ) ) )
2918, 28syl 15 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
( x  e.  ( 
-oo (,) 0 )  \/  x  e.  RR+ )  <->  ( x  <  0  \/  0  <  x ) ) )
3021, 29bitr4d 247 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  =/=  0  <->  (
x  e.  (  -oo (,) 0 )  \/  x  e.  RR+ ) ) )
316, 30mpbid 201 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  (
x  e.  (  -oo (,) 0 )  \/  x  e.  RR+ ) )
32 elun 3329 . . . . . . . 8  |-  ( x  e.  ( (  -oo (,) 0 )  u.  RR+ ) 
<->  ( x  e.  ( 
-oo (,) 0 )  \/  x  e.  RR+ )
)
3331, 32sylibr 203 . . . . . . 7  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  ( (  -oo (,) 0 )  u.  RR+ ) )
3433ex 423 . . . . . 6  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  x  e.  ( (  -oo (,) 0 )  u.  RR+ ) ) )
3534ssrdv 3198 . . . . 5  |-  ( ph  ->  ran  ( RR  _D  F )  C_  (
(  -oo (,) 0 )  u.  RR+ ) )
36 disjssun 3525 . . . . 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 211 . . . 4  |-  ( ph  ->  ( ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0
) )  =  (/)  ->  ran  ( RR  _D  F )  C_  RR+ )
)
3837imp 418 . . 3  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  (  -oo (,) 0 ) )  =  (/) )  ->  ran  ( RR  _D  F )  C_  RR+ )
3910adantr 451 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  A  e.  RR )
4011adantr 451 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  B  e.  RR )
417adantr 451 . . . . 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 5396 . . . . . . . . 9  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
4415, 43mpbid 201 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
45 ffn 5405 . . . . . . . 8  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR  ->  ( RR  _D  F )  Fn  ( A (,) B ) )
4644, 45syl 15 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  Fn  ( A (,) B ) )
4746anim1i 551 . . . . . 6  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( ( RR  _D  F )  Fn  ( A (,) B
)  /\  ran  ( RR 
_D  F )  C_  RR+ ) )
48 df-f 5275 . . . . . 6  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR+  <->  ( ( RR  _D  F )  Fn  ( A (,) B
)  /\  ran  ( RR 
_D  F )  C_  RR+ ) )
4947, 48sylibr 203 . . . . 5  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( RR  _D  F ) : ( A (,) B ) -->
RR+ )
5039, 40, 41, 49dvgt0 19367 . . . 4  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F ) )
5150orcd 381 . . 3  |-  ( (
ph  /\  ran  ( RR 
_D  F )  C_  RR+ )  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
5238, 51syldan 456 . 2  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  (  -oo (,) 0 ) )  =  (/) )  ->  ( F 
Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
53 n0 3477 . . . 4  |-  ( ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0 ) )  =/=  (/)  <->  E. x  x  e.  ( ran  ( RR 
_D  F )  i^i  (  -oo (,) 0
) ) )
54 elin 3371 . . . . . 6  |-  ( x  e.  ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0
) )  <->  ( x  e.  ran  ( RR  _D  F )  /\  x  e.  (  -oo (,) 0
) ) )
55 fvelrnb 5586 . . . . . . . . 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 15 . . . . . . . 8  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  <->  E. y  e.  ( A (,) B ) ( ( RR  _D  F ) `  y
)  =  x ) )
5710adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  A  e.  RR )
5811adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  B  e.  RR )
597adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  F  e.  ( ( A [,] B ) -cn-> RR ) )
6046adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  ( RR  _D  F )  Fn  ( A (,) B ) )
6144adantr 451 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  ( RR  _D  F ) : ( A (,) B ) --> RR )
62 ffvelrn 5679 . . . . . . . . . . . . . . . . 17  |-  ( ( ( RR  _D  F
) : ( A (,) B ) --> RR 
/\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  RR )
6361, 62sylan 457 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  RR )
641ad2antrr 706 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  -.  0  e.  ran  ( RR  _D  F ) )
65 simplrl 736 . . . . . . . . . . . . . . . . . . . . . 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
) )
66 simprl 732 . . . . . . . . . . . . . . . . . . . . . 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
) )
67 ioossicc 10751 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A (,) B )  C_  ( A [,] B )
68 rescncf 18417 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( A (,) B ) 
C_  ( A [,] B )  ->  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  ( F  |`  ( A (,) B
) )  e.  ( ( A (,) B
) -cn-> RR ) ) )
6967, 7, 68mpsyl 59 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( F  |`  ( A (,) B ) )  e.  ( ( A (,) B ) -cn-> RR ) )
7069ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . 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 ) )
71 ax-resscn 8810 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  RR  C_  CC
7271a1i 10 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  RR  C_  CC )
73 fss 5413 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( F : ( A [,] B ) --> RR 
/\  RR  C_  CC )  ->  F : ( A [,] B ) --> CC )
749, 71, 73sylancl 643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  F : ( A [,] B ) --> CC )
7567, 13syl5ss 3203 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( A (,) B
)  C_  RR )
76 eqid 2296 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7776tgioo2 18325 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7876, 77dvres 19277 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
7972, 74, 13, 75, 78syl22anc 1183 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) ) )
80 retop 18286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( topGen ` 
ran  (,) )  e.  Top
81 iooretop 18291 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
82 isopn3i 16835 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( ( topGen `  ran  (,) )  e.  Top  /\  ( A (,) B )  e.  ( topGen `  ran  (,) )
)  ->  ( ( int `  ( topGen `  ran  (,) ) ) `  ( A (,) B ) )  =  ( A (,) B ) )
8380, 81, 82mp2an 653 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( int `  ( topGen ` 
ran  (,) ) ) `  ( A (,) B ) )  =  ( A (,) B )
8483reseq2i 4968 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( RR  _D  F )  |`  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( ( RR 
_D  F )  |`  ( A (,) B ) )
85 fnresdm 5369 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( RR  _D  F )  Fn  ( A (,) B )  ->  (
( RR  _D  F
)  |`  ( A (,) B ) )  =  ( RR  _D  F
) )
8646, 85syl 15 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( ( RR  _D  F )  |`  ( A (,) B ) )  =  ( RR  _D  F ) )
8784, 86syl5eq 2340 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8879, 87eqtrd 2328 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8988dmeqd 4897 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  dom  ( RR  _D  F ) )
9089, 42eqtrd 2328 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( A (,) B ) )
9190ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . 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 ) )
9265, 66, 70, 91dvivth 19373 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
9388ad2antrr 706 . . . . . . . . . . . . . . . . . . . . . . 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 ) )
9493fveq1d 5543 . . . . . . . . . . . . . . . . . . . . . 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
) )
9593fveq1d 5543 . . . . . . . . . . . . . . . . . . . . . 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
) )
9694, 95oveq12d 5892 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
9793rneqd 4922 . . . . . . . . . . . . . . . . . . . . 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 ) )
9892, 96, 973sstr3d 3233 . . . . . . . . . . . . . . . . . . . 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 ) )
9919a1i 10 . . . . . . . . . . . . . . . . . . . . 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 )
100 simplrr 737 . . . . . . . . . . . . . . . . . . . . . . . 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
) )
101 elioomnf 10754 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 0  e.  RR*  ->  ( ( ( RR  _D  F
) `  y )  e.  (  -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 y )  e.  RR  /\  ( ( RR  _D  F ) `
 y )  <  0 ) ) )
10222, 101ax-mp 8 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( RR  _D  F
) `  y )  e.  (  -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 y )  e.  RR  /\  ( ( RR  _D  F ) `
 y )  <  0 ) )
103100, 102sylib 188 . . . . . . . . . . . . . . . . . . . . . . 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 ) )
104103simprd 449 . . . . . . . . . . . . . . . . . . . . . 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 )
105103simpld 445 . . . . . . . . . . . . . . . . . . . . . . 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 )
106 ltle 8926 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( RR  _D  F ) `  y
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  y )  <  0  ->  ( ( RR  _D  F ) `  y
)  <_  0 ) )
107105, 19, 106sylancl 643 . . . . . . . . . . . . . . . . . . . . . 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 ) )
108104, 107mpd 14 . . . . . . . . . . . . . . . . . . . . 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 )
109 simprr 733 . . . . . . . . . . . . . . . . . . . . 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
) )
11066, 63syldan 456 . . . . . . . . . . . . . . . . . . . . . 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 )
111 elicc2 10731 . . . . . . . . . . . . . . . . . . . . . 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 )
) ) )
112105, 110, 111syl2anc 642 . . . . . . . . . . . . . . . . . . . . 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
) ) ) )
11399, 108, 109, 112mpbir3and 1135 . . . . . . . . . . . . . . . . . . . 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 )
) )
11498, 113sseldd 3194 . . . . . . . . . . . . . . . . . . 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 ) )
115114expr 598 . . . . . . . . . . . . . . . . . 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 ) ) )
11664, 115mtod 168 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  -.  0  <_  ( ( RR  _D  F ) `  z
) )
117 ltnle 8918 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( RR  _D  F ) `  z
)  e.  RR  /\  0  e.  RR )  ->  ( ( ( RR 
_D  F ) `  z )  <  0  <->  -.  0  <_  ( ( RR  _D  F ) `  z ) ) )
11863, 19, 117sylancl 643 . . . . . . . . . . . . . . . . 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 )
) )
119116, 118mpbird 223 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  <  0
)
120 elioomnf 10754 . . . . . . . . . . . . . . . . 17  |-  ( 0  e.  RR*  ->  ( ( ( RR  _D  F
) `  z )  e.  (  -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 z )  e.  RR  /\  ( ( RR  _D  F ) `
 z )  <  0 ) ) )
12122, 120ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( ( ( RR  _D  F
) `  z )  e.  (  -oo (,) 0
)  <->  ( ( ( RR  _D  F ) `
 z )  e.  RR  /\  ( ( RR  _D  F ) `
 z )  <  0 ) )
12263, 119, 121sylanbrc 645 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  (
y  e.  ( A (,) B )  /\  ( ( RR  _D  F ) `  y
)  e.  (  -oo (,) 0 ) ) )  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  ( 
-oo (,) 0 ) )
123122ralrimiva 2639 . . . . . . . . . . . . . 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 ) )
124 ffnfv 5701 . . . . . . . . . . . . . 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 ) ) )
12560, 123, 124sylanbrc 645 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  ( RR  _D  F ) : ( A (,) B ) --> (  -oo (,) 0
) )
12657, 58, 59, 125dvlt0 19368 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( y  e.  ( A (,) B
)  /\  ( ( RR  _D  F ) `  y )  e.  ( 
-oo (,) 0 ) ) )  ->  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) )
127126olcd 382 . . . . . . . . . . 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 ) ) )
128127expr 598 . . . . . . . . . 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 ) ) ) )
129 eleq1 2356 . . . . . . . . . . 11  |-  ( ( ( RR  _D  F
) `  y )  =  x  ->  ( ( ( RR  _D  F
) `  y )  e.  (  -oo (,) 0
)  <->  x  e.  (  -oo (,) 0 ) ) )
130129imbi1d 308 . . . . . . . . . 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 ) ) ) ) )
131128, 130syl5ibcom 211 . . . . . . . . 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 ) ) ) ) )
132131rexlimdva 2680 . . . . . . . 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 ) ) ) ) )
13356, 132sylbid 206 . . . . . . 7  |-  ( ph  ->  ( x  e.  ran  ( RR  _D  F
)  ->  ( x  e.  (  -oo (,) 0
)  ->  ( F  Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) ) )
134133imp3a 420 . . . . . 6  |-  ( ph  ->  ( ( x  e. 
ran  ( RR  _D  F )  /\  x  e.  (  -oo (,) 0
) )  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
13554, 134syl5bi 208 . . . . 5  |-  ( ph  ->  ( x  e.  ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0 ) )  ->  ( F  Isom  <  ,  <  ( ( A [,] B ) ,  ran  F )  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
136135exlimdv 1626 . . . 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 ) ) ) )
13753, 136syl5bi 208 . . 3  |-  ( ph  ->  ( ( ran  ( RR  _D  F )  i^i  (  -oo (,) 0
) )  =/=  (/)  ->  ( F  Isom  <  ,  <  ( ( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) ) )
138137imp 418 . 2  |-  ( (
ph  /\  ( ran  ( RR  _D  F
)  i^i  (  -oo (,) 0 ) )  =/=  (/) )  ->  ( F 
Isom  <  ,  <  (
( A [,] B
) ,  ran  F
)  \/  F  Isom  <  ,  `'  <  ( ( A [,] B ) ,  ran  F ) ) )
13952, 138pm2.61dane 2537 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 176    \/ wo 357    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707    Fn wfn 5266   -->wf 5267   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    -oocmnf 8881   RR*cxr 8882    < clt 8883    <_ cle 8884   RR+crp 10370   (,)cioo 10672   [,]cicc 10675   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   Topctop 16647   intcnt 16770   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  dvne0f1  19375  dvcnvrelem1  19380
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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