MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvivth Unicode version

Theorem dvivth 19357
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 18818 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 10712 . . . 4  |-  ( A (,) B )  C_  RR
2 dvivth.1 . . . 4  |-  ( ph  ->  M  e.  ( A (,) B ) )
31, 2sseldi 3178 . . 3  |-  ( ph  ->  M  e.  RR )
4 dvivth.2 . . . 4  |-  ( ph  ->  N  e.  ( A (,) B ) )
51, 4sseldi 3178 . . 3  |-  ( ph  ->  N  e.  RR )
63, 5lttri4d 8960 . 2  |-  ( ph  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
72adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  e.  ( A (,) B ) )
84adantr 451 . . . . . . . . . 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 18397 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  F :
( A (,) B
) --> RR )
119, 10syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( A (,) B ) --> RR )
12 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( F : ( A (,) B ) --> RR 
/\  w  e.  ( A (,) B ) )  ->  ( F `  w )  e.  RR )
1311, 12sylan 457 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  ( F `  w )  e.  RR )
1413renegcld 9210 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  -u ( F `
 w )  e.  RR )
15 eqid 2283 . . . . . . . . . . . . 13  |-  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)
1614, 15fmptd 5684 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR )
17 ax-resscn 8794 . . . . . . . . . . . . 13  |-  RR  C_  CC
18 ssid 3197 . . . . . . . . . . . . . . . 16  |-  CC  C_  CC
19 cncfss 18403 . . . . . . . . . . . . . . . 16  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A (,) B
) -cn-> RR )  C_  (
( A (,) B
) -cn-> CC ) )
2017, 18, 19mp2an 653 . . . . . . . . . . . . . . 15  |-  ( ( A (,) B )
-cn-> RR )  C_  (
( A (,) B
) -cn-> CC )
2120, 9sseldi 3178 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
2215negfcncf 18422 . . . . . . . . . . . . . 14  |-  ( F  e.  ( ( A (,) B ) -cn-> CC )  ->  ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> CC ) )
2321, 22syl 15 . . . . . . . . . . . . 13  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> CC ) )
24 cncffvrn 18402 . . . . . . . . . . . . 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 ) )
2517, 23, 24sylancr 644 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR )  <-> 
( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR ) )
2616, 25mpbird 223 . . . . . . . . . . 11  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR ) )
2726adantr 451 . . . . . . . . . 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 ) )
28 reex 8828 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2928prid1 3734 . . . . . . . . . . . . . 14  |-  RR  e.  { RR ,  CC }
3029a1i 10 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  RR  e.  { RR ,  CC } )
3111adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F : ( A (,) B ) --> RR )
3231, 12sylan 457 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  RR )
3332recnd 8861 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  CC )
34 fvex 5539 . . . . . . . . . . . . . 14  |-  ( ( RR  _D  F ) `
 w )  e. 
_V
3534a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( ( RR  _D  F ) `  w
)  e.  _V )
3631feqmptd 5575 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  =  ( w  e.  ( A (,) B
)  |->  ( F `  w ) ) )
3736oveq2d 5874 . . . . . . . . . . . . . 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 ) ) ) )
38 dvfre 19300 . . . . . . . . . . . . . . . . . 18  |-  ( ( F : ( A (,) B ) --> RR 
/\  ( A (,) B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
3911, 1, 38sylancl 643 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
40 dvivth.4 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
4140feq2d 5380 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
4239, 41mpbid 201 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
4342adantr 451 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
) : ( A (,) B ) --> RR )
4443feqmptd 5575 . . . . . . . . . . . . . 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 ) ) )
4537, 44eqtr3d 2317 . . . . . . . . . . . . 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 )
) )
4630, 33, 35, 45dvmptneg 19315 . . . . . . . . . . . 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
) ) )
4746dmeqd 4881 . . . . . . . . . . 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
) ) )
48 dmmptg 5170 . . . . . . . . . . . 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 ) )
49 negex 9050 . . . . . . . . . . . . 13  |-  -u (
( RR  _D  F
) `  w )  e.  _V
5049a1i 10 . . . . . . . . . . . 12  |-  ( w  e.  ( A (,) B )  ->  -u (
( RR  _D  F
) `  w )  e.  _V )
5148, 50mprg 2612 . . . . . . . . . . 11  |-  dom  (
w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) )  =  ( A (,) B )
5247, 51syl6eq 2331 . . . . . . . . . 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 ) )
53 simprl 732 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  <  N )
54 simprr 733 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) ) )
55 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  F
) : ( A (,) B ) --> RR 
/\  M  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  M )  e.  RR )
5642, 2, 55syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  _D  F ) `  M
)  e.  RR )
5756adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  F ) `  M
)  e.  RR )
584, 40eleqtrrd 2360 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  dom  ( RR  _D  F ) )
59 ffvelrn 5663 . . . . . . . . . . . . . . 15  |-  ( ( ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR 
/\  N  e.  dom  ( RR  _D  F
) )  ->  (
( RR  _D  F
) `  N )  e.  RR )
6039, 58, 59syl2anc 642 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR )
6160adantr 451 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  F ) `  N
)  e.  RR )
62 iccssre 10731 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( RR  _D  F ) `  M
)  e.  RR  /\  ( ( RR  _D  F ) `  N
)  e.  RR )  ->  ( ( ( RR  _D  F ) `
 M ) [,] ( ( RR  _D  F ) `  N
) )  C_  RR )
6356, 60, 62syl2anc 642 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  RR )
6463adantr 451 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  RR )
6564, 54sseldd 3181 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  RR )
66 iccneg 10757 . . . . . . . . . . . . 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 )
) ) )
6757, 61, 65, 66syl3anc 1182 . . . . . . . . . . . 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 )
) ) )
6854, 67mpbid 201 . . . . . . . . . . 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 )
) )
6946fveq1d 5527 . . . . . . . . . . . . 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
) )
70 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( w  =  N  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  N ) )
7170negeqd 9046 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  N ) )
72 eqid 2283 . . . . . . . . . . . . . . 15  |-  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)
73 negex 9050 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  N )  e.  _V
7471, 72, 73fvmpt 5602 . . . . . . . . . . . . . 14  |-  ( N  e.  ( A (,) B )  ->  (
( w  e.  ( A (,) B ) 
|->  -u ( ( RR 
_D  F ) `  w ) ) `  N )  =  -u ( ( RR  _D  F ) `  N
) )
758, 74syl 15 . . . . . . . . . . . . 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
) )
7669, 75eqtrd 2315 . . . . . . . . . . . 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 )
)
7746fveq1d 5527 . . . . . . . . . . . . 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
) )
78 fveq2 5525 . . . . . . . . . . . . . . . 16  |-  ( w  =  M  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  M ) )
7978negeqd 9046 . . . . . . . . . . . . . . 15  |-  ( w  =  M  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  M ) )
80 negex 9050 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  M )  e.  _V
8179, 72, 80fvmpt 5602 . . . . . . . . . . . . . 14  |-  ( M  e.  ( A (,) B )  ->  (
( w  e.  ( A (,) B ) 
|->  -u ( ( RR 
_D  F ) `  w ) ) `  M )  =  -u ( ( RR  _D  F ) `  M
) )
827, 81syl 15 . . . . . . . . . . . . 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
) )
8377, 82eqtrd 2315 . . . . . . . . . . . 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 )
)
8476, 83oveq12d 5876 . . . . . . . . . . 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 )
) )
8568, 84eleqtrrd 2360 . . . . . . . . . 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
) ) )
86 eqid 2283 . . . . . . . . . 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 ) ) )
877, 8, 27, 52, 53, 85, 86dvivthlem2 19356 . . . . . . . . 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 ) ) ) )
8846rneqd 4906 . . . . . . . . 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
) ) )
8987, 88eleqtrd 2359 . . . . . . . 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
) ) )
90 negex 9050 . . . . . . . . 9  |-  -u x  e.  _V
9172elrnmpt 4926 . . . . . . . . 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
) ) )
9290, 91ax-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
) )
9389, 92sylib 188 . . . . . . 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 ) )
9465recnd 8861 . . . . . . . . . . 11  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  CC )
9594adantr 451 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  ->  x  e.  CC )
9630, 33, 35, 45dvmptcl 19308 . . . . . . . . . 10  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( ( RR  _D  F ) `  w
)  e.  CC )
97 neg11 9098 . . . . . . . . . 10  |-  ( ( x  e.  CC  /\  ( ( RR  _D  F ) `  w
)  e.  CC )  ->  ( -u x  =  -u ( ( RR 
_D  F ) `  w )  <->  x  =  ( ( RR  _D  F ) `  w
) ) )
9895, 96, 97syl2anc 642 . . . . . . . . 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 )
) )
99 eqcom 2285 . . . . . . . . 9  |-  ( x  =  ( ( RR 
_D  F ) `  w )  <->  ( ( RR  _D  F ) `  w )  =  x )
10098, 99syl6bb 252 . . . . . . . 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 ) )
101100rexbidva 2560 . . . . . . 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 ) )
10293, 101mpbid 201 . . . . . 6  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  E. w  e.  ( A (,) B ) ( ( RR  _D  F
) `  w )  =  x )
103 ffn 5389 . . . . . . . 8  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR  ->  ( RR  _D  F )  Fn  ( A (,) B ) )
10443, 103syl 15 . . . . . . 7  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
)  Fn  ( A (,) B ) )
105 fvelrnb 5570 . . . . . . 7  |-  ( ( RR  _D  F )  Fn  ( A (,) B )  ->  (
x  e.  ran  ( RR  _D  F )  <->  E. w  e.  ( A (,) B
) ( ( RR 
_D  F ) `  w )  =  x ) )
106104, 105syl 15 . . . . . 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 ) )
107102, 106mpbird 223 . . . . 5  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ran  ( RR 
_D  F ) )
108107expr 598 . . . 4  |-  ( (
ph  /\  M  <  N )  ->  ( x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  ->  x  e.  ran  ( RR  _D  F
) ) )
109108ssrdv 3185 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
110 fveq2 5525 . . . . . 6  |-  ( M  =  N  ->  (
( RR  _D  F
) `  M )  =  ( ( RR 
_D  F ) `  N ) )
111110oveq1d 5873 . . . . 5  |-  ( M  =  N  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  ( ( ( RR  _D  F ) `
 N ) [,] ( ( RR  _D  F ) `  N
) ) )
11260rexrd 8881 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR* )
113 iccid 10701 . . . . . 6  |-  ( ( ( RR  _D  F
) `  N )  e.  RR*  ->  ( (
( RR  _D  F
) `  N ) [,] ( ( RR  _D  F ) `  N
) )  =  {
( ( RR  _D  F ) `  N
) } )
114112, 113syl 15 . . . . 5  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  N ) [,] (
( RR  _D  F
) `  N )
)  =  { ( ( RR  _D  F
) `  N ) } )
115111, 114sylan9eqr 2337 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  { ( ( RR  _D  F ) `
 N ) } )
116 ffn 5389 . . . . . . . 8  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> RR 
->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
11739, 116syl 15 . . . . . . 7  |-  ( ph  ->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
118 fnfvelrn 5662 . . . . . . 7  |-  ( ( ( RR  _D  F
)  Fn  dom  ( RR  _D  F )  /\  N  e.  dom  ( RR 
_D  F ) )  ->  ( ( RR 
_D  F ) `  N )  e.  ran  ( RR  _D  F
) )
119117, 58, 118syl2anc 642 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  ran  ( RR  _D  F ) )
120119snssd 3760 . . . . 5  |-  ( ph  ->  { ( ( RR 
_D  F ) `  N ) }  C_  ran  ( RR  _D  F
) )
121120adantr 451 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  { ( ( RR  _D  F
) `  N ) }  C_  ran  ( RR 
_D  F ) )
122115, 121eqsstrd 3212 . . 3  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) ) 
C_  ran  ( RR  _D  F ) )
1234adantr 451 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  e.  ( A (,) B ) )
1242adantr 451 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  e.  ( A (,) B ) )
1259adantr 451 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  e.  ( ( A (,) B ) -cn-> RR ) )
12640adantr 451 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  dom  ( RR  _D  F
)  =  ( A (,) B ) )
127 simprl 732 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  <  M )
128 simprr 733 . . . . . 6  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) ) )
129 eqid 2283 . . . . . 6  |-  ( y  e.  ( A (,) B )  |->  ( ( F `  y )  -  ( x  x.  y ) ) )  =  ( y  e.  ( A (,) B
)  |->  ( ( F `
 y )  -  ( x  x.  y
) ) )
130123, 124, 125, 126, 127, 128, 129dvivthlem2 19356 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ran  ( RR 
_D  F ) )
131130expr 598 . . . 4  |-  ( (
ph  /\  N  <  M )  ->  ( x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  ->  x  e.  ran  ( RR  _D  F
) ) )
132131ssrdv 3185 . . 3  |-  ( (
ph  /\  N  <  M )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
133109, 122, 1323jaodan 1248 . 2  |-  ( (
ph  /\  ( M  <  N  \/  M  =  N  \/  N  < 
M ) )  -> 
( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  ran  ( RR 
_D  F ) )
1346, 133mpdan 649 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 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   E.wrex 2544   _Vcvv 2788    C_ wss 3152   {csn 3640   {cpr 3641   class class class wbr 4023    e. cmpt 4077   dom cdm 4689   ran crn 4690    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736    x. cmul 8742   RR*cxr 8866    < clt 8867    - cmin 9037   -ucneg 9038   (,)cioo 10656   [,]cicc 10659   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  dvne0  19358
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-ico 10662  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-lp 16868  df-perf 16869  df-cn 16957  df-cnp 16958  df-haus 17043  df-cmp 17114  df-tx 17257  df-hmeo 17446  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635  df-xms 17885  df-ms 17886  df-tms 17887  df-cncf 18382  df-limc 19216  df-dv 19217
  Copyright terms: Public domain W3C validator