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Theorem dvne0 22280
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 2539 . . . . . . . . . . . . 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 2680 . . . . . . . . . 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 21265 . . . . . . . . . . . . . . 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 11618 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1310, 11, 12syl2anc 661 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A [,] B
)  C_  RR )
14 dvfre 22222 . . . . . . . . . . . . . 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 3509 . . . . . . . . . . 11  |-  ( (
ph  /\  x  e.  ran  ( RR  _D  F
) )  ->  x  e.  RR )
19 0re 9608 . . . . . . . . . . 11  |-  0  e.  RR
20 lttri2 9679 . . . . . . . . . . 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 9652 . . . . . . . . . . . . . 14  |-  0  e.  RR*
23 elioomnf 11631 . . . . . . . . . . . . . 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 901 . . . . . . . . . . . 12  |-  ( x  e.  RR  ->  (
x  e.  ( -oo (,) 0 )  <->  x  <  0 ) )
26 elrp 11234 . . . . . . . . . . . . 13  |-  ( x  e.  RR+  <->  ( x  e.  RR  /\  0  < 
x ) )
2726baib 901 . . . . . . . . . . . 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 3650 . . . . . . . 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 3515 . . . . 5  |-  ( ph  ->  ran  ( RR  _D  F )  C_  (
( -oo (,) 0 )  u.  RR+ ) )
36 disjssun 3889 . . . . 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 22273 . . . 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 3799 . . . 4  |-  ( ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0 ) )  =/=  (/)  <->  E. x  x  e.  ( ran  ( RR 
_D  F )  i^i  ( -oo (,) 0
) ) )
54 elin 3692 . . . . . 6  |-  ( x  e.  ( ran  ( RR  _D  F )  i^i  ( -oo (,) 0
) )  <->  ( x  e.  ran  ( RR  _D  F )  /\  x  e.  ( -oo (,) 0
) ) )
55 fvelrnb 5921 . . . . . . . . 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 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 11622 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( A (,) B )  C_  ( A [,] B )
67 rescncf 21269 . . . . . . . . . . . . . . . . . . . . . . . 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 9561 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( A (,) B
)  C_  RR )
75 eqid 2467 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
7675tgioo2 21176 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
7775, 76dvres 22183 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 21136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( topGen ` 
ran  (,) )  e.  Top
80 iooretop 21141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( A (,) B )  e.  ( topGen `  ran  (,) )
81 isopn3i 19451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5276 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2520 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( RR  _D  F )  |`  (
( int `  ( topGen `
 ran  (,) )
) `  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8778, 86eqtrd 2508 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  ( RR  _D  F ) )
8887dmeqd 5211 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  dom  ( RR  _D  ( F  |`  ( A (,) B ) ) )  =  dom  ( RR  _D  F ) )
8988, 42eqtrd 2508 . . . . . . . . . . . . . . . . . . . . . . 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 22279 . . . . . . . . . . . . . . . . . . . . 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 6313 . . . . . . . . . . . . . . . . . . . . 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 5236 . . . . . . . . . . . . . . . . . . . . 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 3551 . . . . . . . . . . . . . . . . . . . 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 11631 . . . . . . . . . . . . . . . . . . . . . . . . 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 9685 . . . . . . . . . . . . . . . . . . . . . . 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 11601 . . . . . . . . . . . . . . . . . . . . . 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 3510 . . . . . . . . . . . . . . . . . . 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 9676 . . . . . . . . . . . . . . . . . 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 11631 . . . . . . . . . . . . . . . . 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 2881 . . . . . . . . . . . . . 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 22274 . . . . . . . . . . . 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 2539 . . . . . . . . . . 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 2959 . . . . . . . 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 1700 . . . 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 2785 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2817   E.wrex 2818    u. cun 3479    i^i cin 3480    C_ wss 3481   (/)c0 3790   class class class wbr 4453   `'ccnv 5004   dom cdm 5005   ran crn 5006    |` cres 5007    Fn wfn 5589   -->wf 5590   ` cfv 5594    Isom wiso 5595  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504   -oocmnf 9638   RR*cxr 9639    < clt 9640    <_ cle 9641   RR+crp 11232   (,)cioo 11541   [,]cicc 11544   TopOpenctopn 14694   topGenctg 14710  ℂfldccnfld 18290   Topctop 19263   intcnt 19386   -cn->ccncf 21248    _D cdv 22135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-supp 6914  df-recs 7054  df-rdg 7088  df-1o 7142  df-2o 7143  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-ixp 7482  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-fsupp 7842  df-fi 7883  df-sup 7913  df-oi 7947  df-card 8332  df-cda 8560  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-n0 10808  df-z 10877  df-dec 10989  df-uz 11095  df-q 11195  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ioo 11545  df-ico 11547  df-icc 11548  df-fz 11685  df-fzo 11805  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-struct 14509  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-ress 14514  df-plusg 14585  df-mulr 14586  df-starv 14587  df-sca 14588  df-vsca 14589  df-ip 14590  df-tset 14591  df-ple 14592  df-ds 14594  df-unif 14595  df-hom 14596  df-cco 14597  df-rest 14695  df-topn 14696  df-0g 14714  df-gsum 14715  df-topgen 14716  df-pt 14717  df-prds 14720  df-xrs 14774  df-qtop 14779  df-imas 14780  df-xps 14782  df-mre 14858  df-mrc 14859  df-acs 14861  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-submnd 15840  df-mulg 15932  df-cntz 16227  df-cmn 16673  df-psmet 18281  df-xmet 18282  df-met 18283  df-bl 18284  df-mopn 18285  df-fbas 18286  df-fg 18287  df-cnfld 18291  df-top 19268  df-bases 19270  df-topon 19271  df-topsp 19272  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-lp 19505  df-perf 19506  df-cn 19596  df-cnp 19597  df-haus 19684  df-cmp 19755  df-tx 19931  df-hmeo 20124  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309  df-xms 20691  df-ms 20692  df-tms 20693  df-cncf 21250  df-limc 22138  df-dv 22139
This theorem is referenced by:  dvne0f1  22281  dvcnvrelem1  22286
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