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Theorem dvivth 19882
Description: Darboux' theorem, or the intermediate value theorem for derivatives. A differentiable function's derivative satisfies the intermediate value property, even though it may not be continuous (so that ivthicc 19343 does not directly apply). (Contributed by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
dvivth.1  |-  ( ph  ->  M  e.  ( A (,) B ) )
dvivth.2  |-  ( ph  ->  N  e.  ( A (,) B ) )
dvivth.3  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> RR ) )
dvivth.4  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
Assertion
Ref Expression
dvivth  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  ran  ( RR 
_D  F ) )

Proof of Theorem dvivth
Dummy variables  x  w  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ioossre 10961 . . . 4  |-  ( A (,) B )  C_  RR
2 dvivth.1 . . . 4  |-  ( ph  ->  M  e.  ( A (,) B ) )
31, 2sseldi 3338 . . 3  |-  ( ph  ->  M  e.  RR )
4 dvivth.2 . . . 4  |-  ( ph  ->  N  e.  ( A (,) B ) )
51, 4sseldi 3338 . . 3  |-  ( ph  ->  N  e.  RR )
63, 5lttri4d 9203 . 2  |-  ( ph  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
72adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  e.  ( A (,) B ) )
84adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  e.  ( A (,) B ) )
9 dvivth.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> RR ) )
10 cncff 18911 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  F :
( A (,) B
) --> RR )
119, 10syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( A (,) B ) --> RR )
1211ffvelrnda 5861 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  ( F `  w )  e.  RR )
1312renegcld 9453 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  -u ( F `
 w )  e.  RR )
14 eqid 2435 . . . . . . . . . . . . 13  |-  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)
1513, 14fmptd 5884 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR )
16 ax-resscn 9036 . . . . . . . . . . . . 13  |-  RR  C_  CC
17 ssid 3359 . . . . . . . . . . . . . . . 16  |-  CC  C_  CC
18 cncfss 18917 . . . . . . . . . . . . . . . 16  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A (,) B
) -cn-> RR )  C_  (
( A (,) B
) -cn-> CC ) )
1916, 17, 18mp2an 654 . . . . . . . . . . . . . . 15  |-  ( ( A (,) B )
-cn-> RR )  C_  (
( A (,) B
) -cn-> CC )
2019, 9sseldi 3338 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
2114negfcncf 18937 . . . . . . . . . . . . . 14  |-  ( F  e.  ( ( A (,) B ) -cn-> CC )  ->  ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> CC ) )
2220, 21syl 16 . . . . . . . . . . . . 13  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> CC ) )
23 cncffvrn 18916 . . . . . . . . . . . . 13  |-  ( ( RR  C_  CC  /\  (
w  e.  ( A (,) B )  |->  -u ( F `  w ) )  e.  ( ( A (,) B )
-cn-> CC ) )  -> 
( ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR )  <-> 
( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR ) )
2416, 22, 23sylancr 645 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR )  <-> 
( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR ) )
2515, 24mpbird 224 . . . . . . . . . . 11  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR ) )
2625adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR ) )
27 reex 9070 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2827prid1 3904 . . . . . . . . . . . . . 14  |-  RR  e.  { RR ,  CC }
2928a1i 11 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  RR  e.  { RR ,  CC } )
3011adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F : ( A (,) B ) --> RR )
3130ffvelrnda 5861 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  RR )
3231recnd 9103 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  CC )
33 fvex 5733 . . . . . . . . . . . . . 14  |-  ( ( RR  _D  F ) `
 w )  e. 
_V
3433a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( ( RR  _D  F ) `  w
)  e.  _V )
3530feqmptd 5770 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  =  ( w  e.  ( A (,) B
)  |->  ( F `  w ) ) )
3635oveq2d 6088 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
)  =  ( RR 
_D  ( w  e.  ( A (,) B
)  |->  ( F `  w ) ) ) )
37 dvfre 19825 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : ( A (,) B ) --> RR 
/\  ( A (,) B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
3811, 1, 37sylancl 644 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
39 dvivth.4 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
4039feq2d 5572 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
4138, 40mpbid 202 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
4241adantr 452 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
) : ( A (,) B ) --> RR )
4342feqmptd 5770 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
)  =  ( w  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 w ) ) )
4436, 43eqtr3d 2469 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  (
w  e.  ( A (,) B )  |->  ( F `  w ) ) )  =  ( w  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  w )
) )
4529, 32, 34, 44dvmptneg 19840 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  (
w  e.  ( A (,) B )  |->  -u ( F `  w ) ) )  =  ( w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) ) )
4645dmeqd 5063 . . . . . . . . . . 11  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  dom  ( RR  _D  (
w  e.  ( A (,) B )  |->  -u ( F `  w ) ) )  =  dom  ( w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) ) )
47 dmmptg 5358 . . . . . . . . . . . 12  |-  ( A. w  e.  ( A (,) B ) -u (
( RR  _D  F
) `  w )  e.  _V  ->  dom  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  =  ( A (,) B ) )
48 negex 9293 . . . . . . . . . . . . 13  |-  -u (
( RR  _D  F
) `  w )  e.  _V
4948a1i 11 . . . . . . . . . . . 12  |-  ( w  e.  ( A (,) B )  ->  -u (
( RR  _D  F
) `  w )  e.  _V )
5047, 49mprg 2767 . . . . . . . . . . 11  |-  dom  (
w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) )  =  ( A (,) B )
5146, 50syl6eq 2483 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  dom  ( RR  _D  (
w  e.  ( A (,) B )  |->  -u ( F `  w ) ) )  =  ( A (,) B ) )
52 simprl 733 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  <  N )
53 simprr 734 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) ) )
5441, 2ffvelrnd 5862 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  _D  F ) `  M
)  e.  RR )
5554adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  F ) `  M
)  e.  RR )
564, 39eleqtrrd 2512 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  dom  ( RR  _D  F ) )
5738, 56ffvelrnd 5862 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR )
5857adantr 452 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  F ) `  N
)  e.  RR )
59 iccssre 10981 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( RR  _D  F ) `  M
)  e.  RR  /\  ( ( RR  _D  F ) `  N
)  e.  RR )  ->  ( ( ( RR  _D  F ) `
 M ) [,] ( ( RR  _D  F ) `  N
) )  C_  RR )
6054, 57, 59syl2anc 643 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  RR )
6160adantr 452 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  RR )
6261, 53sseldd 3341 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  RR )
63 iccneg 11007 . . . . . . . . . . . . 13  |-  ( ( ( ( RR  _D  F ) `  M
)  e.  RR  /\  ( ( RR  _D  F ) `  N
)  e.  RR  /\  x  e.  RR )  ->  ( x  e.  ( ( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  <->  -u x  e.  ( -u ( ( RR  _D  F ) `  N
) [,] -u (
( RR  _D  F
) `  M )
) ) )
6455, 58, 62, 63syl3anc 1184 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( x  e.  ( ( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  <->  -u x  e.  ( -u ( ( RR  _D  F ) `  N
) [,] -u (
( RR  _D  F
) `  M )
) ) )
6553, 64mpbid 202 . . . . . . . . . . 11  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  -u x  e.  ( -u ( ( RR  _D  F ) `  N
) [,] -u (
( RR  _D  F
) `  M )
) )
6645fveq1d 5721 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  ( w  e.  ( A (,) B )  |->  -u ( F `  w ) ) ) `  N
)  =  ( ( w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) ) `  N
) )
67 fveq2 5719 . . . . . . . . . . . . . . . 16  |-  ( w  =  N  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  N ) )
6867negeqd 9289 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  N ) )
69 eqid 2435 . . . . . . . . . . . . . . 15  |-  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)
70 negex 9293 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  N )  e.  _V
7168, 69, 70fvmpt 5797 . . . . . . . . . . . . . 14  |-  ( N  e.  ( A (,) B )  ->  (
( w  e.  ( A (,) B ) 
|->  -u ( ( RR 
_D  F ) `  w ) ) `  N )  =  -u ( ( RR  _D  F ) `  N
) )
728, 71syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( w  e.  ( A (,) B
)  |->  -u ( ( RR 
_D  F ) `  w ) ) `  N )  =  -u ( ( RR  _D  F ) `  N
) )
7366, 72eqtrd 2467 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  ( w  e.  ( A (,) B )  |->  -u ( F `  w ) ) ) `  N
)  =  -u (
( RR  _D  F
) `  N )
)
7445fveq1d 5721 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  ( w  e.  ( A (,) B )  |->  -u ( F `  w ) ) ) `  M
)  =  ( ( w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) ) `  M
) )
75 fveq2 5719 . . . . . . . . . . . . . . . 16  |-  ( w  =  M  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  M ) )
7675negeqd 9289 . . . . . . . . . . . . . . 15  |-  ( w  =  M  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  M ) )
77 negex 9293 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  M )  e.  _V
7876, 69, 77fvmpt 5797 . . . . . . . . . . . . . 14  |-  ( M  e.  ( A (,) B )  ->  (
( w  e.  ( A (,) B ) 
|->  -u ( ( RR 
_D  F ) `  w ) ) `  M )  =  -u ( ( RR  _D  F ) `  M
) )
797, 78syl 16 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( w  e.  ( A (,) B
)  |->  -u ( ( RR 
_D  F ) `  w ) ) `  M )  =  -u ( ( RR  _D  F ) `  M
) )
8074, 79eqtrd 2467 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  ( w  e.  ( A (,) B )  |->  -u ( F `  w ) ) ) `  M
)  =  -u (
( RR  _D  F
) `  M )
)
8173, 80oveq12d 6090 . . . . . . . . . . 11  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( ( RR 
_D  ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) ) ) `
 N ) [,] ( ( RR  _D  ( w  e.  ( A (,) B )  |->  -u ( F `  w ) ) ) `  M
) )  =  (
-u ( ( RR 
_D  F ) `  N ) [,] -u (
( RR  _D  F
) `  M )
) )
8265, 81eleqtrrd 2512 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  -u x  e.  ( ( ( RR  _D  (
w  e.  ( A (,) B )  |->  -u ( F `  w ) ) ) `  N
) [,] ( ( RR  _D  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
) ) `  M
) ) )
83 eqid 2435 . . . . . . . . . 10  |-  ( y  e.  ( A (,) B )  |->  ( ( ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) `  y )  -  ( -u x  x.  y ) ) )  =  ( y  e.  ( A (,) B )  |->  ( ( ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) ) `  y )  -  ( -u x  x.  y ) ) )
847, 8, 26, 51, 52, 82, 83dvivthlem2 19881 . . . . . . . . 9  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  -u x  e.  ran  ( RR  _D  ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) ) ) )
8545rneqd 5088 . . . . . . . . 9  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  ran  ( RR  _D  (
w  e.  ( A (,) B )  |->  -u ( F `  w ) ) )  =  ran  ( w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) ) )
8684, 85eleqtrd 2511 . . . . . . . 8  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  -u x  e.  ran  (
w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) ) )
87 negex 9293 . . . . . . . . 9  |-  -u x  e.  _V
8869elrnmpt 5108 . . . . . . . . 9  |-  ( -u x  e.  _V  ->  (
-u x  e.  ran  ( w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) )  <->  E. w  e.  ( A (,) B
) -u x  =  -u ( ( RR  _D  F ) `  w
) ) )
8987, 88ax-mp 8 . . . . . . . 8  |-  ( -u x  e.  ran  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  <->  E. w  e.  ( A (,) B )
-u x  =  -u ( ( RR  _D  F ) `  w
) )
9086, 89sylib 189 . . . . . . 7  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  E. w  e.  ( A (,) B ) -u x  =  -u ( ( RR  _D  F ) `
 w ) )
9162recnd 9103 . . . . . . . . . . 11  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  CC )
9291adantr 452 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  ->  x  e.  CC )
9329, 32, 34, 44dvmptcl 19833 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( ( RR  _D  F ) `  w
)  e.  CC )
9492, 93neg11ad 9396 . . . . . . . . 9  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( -u x  =  -u ( ( RR  _D  F ) `  w
)  <->  x  =  (
( RR  _D  F
) `  w )
) )
95 eqcom 2437 . . . . . . . . 9  |-  ( x  =  ( ( RR 
_D  F ) `  w )  <->  ( ( RR  _D  F ) `  w )  =  x )
9694, 95syl6bb 253 . . . . . . . 8  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( -u x  =  -u ( ( RR  _D  F ) `  w
)  <->  ( ( RR 
_D  F ) `  w )  =  x ) )
9796rexbidva 2714 . . . . . . 7  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( E. w  e.  ( A (,) B
) -u x  =  -u ( ( RR  _D  F ) `  w
)  <->  E. w  e.  ( A (,) B ) ( ( RR  _D  F ) `  w
)  =  x ) )
9890, 97mpbid 202 . . . . . 6  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  E. w  e.  ( A (,) B ) ( ( RR  _D  F
) `  w )  =  x )
99 ffn 5582 . . . . . . . 8  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR  ->  ( RR  _D  F )  Fn  ( A (,) B ) )
10042, 99syl 16 . . . . . . 7  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
)  Fn  ( A (,) B ) )
101 fvelrnb 5765 . . . . . . 7  |-  ( ( RR  _D  F )  Fn  ( A (,) B )  ->  (
x  e.  ran  ( RR  _D  F )  <->  E. w  e.  ( A (,) B
) ( ( RR 
_D  F ) `  w )  =  x ) )
102100, 101syl 16 . . . . . 6  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( x  e.  ran  ( RR  _D  F
)  <->  E. w  e.  ( A (,) B ) ( ( RR  _D  F ) `  w
)  =  x ) )
10398, 102mpbird 224 . . . . 5  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ran  ( RR 
_D  F ) )
104103expr 599 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  ( x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  ->  x  e.  ran  ( RR  _D  F
) ) )
105104ssrdv 3346 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
106 fveq2 5719 . . . . . 6  |-  ( M  =  N  ->  (
( RR  _D  F
) `  M )  =  ( ( RR 
_D  F ) `  N ) )
107106oveq1d 6087 . . . . 5  |-  ( M  =  N  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  ( ( ( RR  _D  F ) `
 N ) [,] ( ( RR  _D  F ) `  N
) ) )
10857rexrd 9123 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR* )
109 iccid 10950 . . . . . 6  |-  ( ( ( RR  _D  F
) `  N )  e.  RR*  ->  ( (
( RR  _D  F
) `  N ) [,] ( ( RR  _D  F ) `  N
) )  =  {
( ( RR  _D  F ) `  N
) } )
110108, 109syl 16 . . . . 5  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  N ) [,] (
( RR  _D  F
) `  N )
)  =  { ( ( RR  _D  F
) `  N ) } )
111107, 110sylan9eqr 2489 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  { ( ( RR  _D  F ) `
 N ) } )
112 ffn 5582 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> RR 
->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
11338, 112syl 16 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
114 fnfvelrn 5858 . . . . . . 7  |-  ( ( ( RR  _D  F
)  Fn  dom  ( RR  _D  F )  /\  N  e.  dom  ( RR 
_D  F ) )  ->  ( ( RR 
_D  F ) `  N )  e.  ran  ( RR  _D  F
) )
115113, 56, 114syl2anc 643 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  ran  ( RR  _D  F ) )
116115snssd 3935 . . . . 5  |-  ( ph  ->  { ( ( RR 
_D  F ) `  N ) }  C_  ran  ( RR  _D  F
) )
117116adantr 452 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  { ( ( RR  _D  F
) `  N ) }  C_  ran  ( RR 
_D  F ) )
118111, 117eqsstrd 3374 . . 3  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) ) 
C_  ran  ( RR  _D  F ) )
1194adantr 452 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  e.  ( A (,) B ) )
1202adantr 452 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  e.  ( A (,) B ) )
1219adantr 452 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  e.  ( ( A (,) B ) -cn-> RR ) )
12239adantr 452 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  dom  ( RR  _D  F
)  =  ( A (,) B ) )
123 simprl 733 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  <  M )
124 simprr 734 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) ) )
125 eqid 2435 . . . . . 6  |-  ( y  e.  ( A (,) B )  |->  ( ( F `  y )  -  ( x  x.  y ) ) )  =  ( y  e.  ( A (,) B
)  |->  ( ( F `
 y )  -  ( x  x.  y
) ) )
126119, 120, 121, 122, 123, 124, 125dvivthlem2 19881 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ran  ( RR 
_D  F ) )
127126expr 599 . . . 4  |-  ( (
ph  /\  N  <  M )  ->  ( x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  ->  x  e.  ran  ( RR  _D  F
) ) )
128127ssrdv 3346 . . 3  |-  ( (
ph  /\  N  <  M )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
129105, 118, 1283jaodan 1250 . 2  |-  ( (
ph  /\  ( M  <  N  \/  M  =  N  \/  N  < 
M ) )  -> 
( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  ran  ( RR 
_D  F ) )
1306, 129mpdan 650 1  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  ran  ( RR 
_D  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725   E.wrex 2698   _Vcvv 2948    C_ wss 3312   {csn 3806   {cpr 3807   class class class wbr 4204    e. cmpt 4258   dom cdm 4869   ran crn 4870    Fn wfn 5440   -->wf 5441   ` cfv 5445  (class class class)co 6072   CCcc 8977   RRcr 8978    x. cmul 8984   RR*cxr 9108    < clt 9109    - cmin 9280   -ucneg 9281   (,)cioo 10905   [,]cicc 10908   -cn->ccncf 18894    _D cdv 19738
This theorem is referenced by:  dvne0  19883
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 4692  ax-inf2 7585  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056  ax-pre-sup 9057  ax-addf 9058  ax-mulf 9059
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 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-isom 5454  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-of 6296  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-2o 6716  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-ixp 7055  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-fi 7407  df-sup 7437  df-oi 7468  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-div 9667  df-nn 9990  df-2 10047  df-3 10048  df-4 10049  df-5 10050  df-6 10051  df-7 10052  df-8 10053  df-9 10054  df-10 10055  df-n0 10211  df-z 10272  df-dec 10372  df-uz 10478  df-q 10564  df-rp 10602  df-xneg 10699  df-xadd 10700  df-xmul 10701  df-ioo 10909  df-ico 10911  df-icc 10912  df-fz 11033  df-fzo 11124  df-seq 11312  df-exp 11371  df-hash 11607  df-cj 11892  df-re 11893  df-im 11894  df-sqr 12028  df-abs 12029  df-struct 13459  df-ndx 13460  df-slot 13461  df-base 13462  df-sets 13463  df-ress 13464  df-plusg 13530  df-mulr 13531  df-starv 13532  df-sca 13533  df-vsca 13534  df-tset 13536  df-ple 13537  df-ds 13539  df-unif 13540  df-hom 13541  df-cco 13542  df-rest 13638  df-topn 13639  df-topgen 13655  df-pt 13656  df-prds 13659  df-xrs 13714  df-0g 13715  df-gsum 13716  df-qtop 13721  df-imas 13722  df-xps 13724  df-mre 13799  df-mrc 13800  df-acs 13802  df-mnd 14678  df-submnd 14727  df-mulg 14803  df-cntz 15104  df-cmn 15402  df-psmet 16682  df-xmet 16683  df-met 16684  df-bl 16685  df-mopn 16686  df-fbas 16687  df-fg 16688  df-cnfld 16692  df-top 16951  df-bases 16953  df-topon 16954  df-topsp 16955  df-cld 17071  df-ntr 17072  df-cls 17073  df-nei 17150  df-lp 17188  df-perf 17189  df-cn 17279  df-cnp 17280  df-haus 17367  df-cmp 17438  df-tx 17582  df-hmeo 17775  df-fil 17866  df-fm 17958  df-flim 17959  df-flf 17960  df-xms 18338  df-ms 18339  df-tms 18340  df-cncf 18896  df-limc 19741  df-dv 19742
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