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Theorem dvivth 19847
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 19308 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 10928 . . . 4  |-  ( A (,) B )  C_  RR
2 dvivth.1 . . . 4  |-  ( ph  ->  M  e.  ( A (,) B ) )
31, 2sseldi 3306 . . 3  |-  ( ph  ->  M  e.  RR )
4 dvivth.2 . . . 4  |-  ( ph  ->  N  e.  ( A (,) B ) )
51, 4sseldi 3306 . . 3  |-  ( ph  ->  N  e.  RR )
63, 5lttri4d 9170 . 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 18876 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  F :
( A (,) B
) --> RR )
119, 10syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( A (,) B ) --> RR )
1211ffvelrnda 5829 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  ( F `  w )  e.  RR )
1312renegcld 9420 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  -u ( F `
 w )  e.  RR )
14 eqid 2404 . . . . . . . . . . . . 13  |-  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)
1513, 14fmptd 5852 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR )
16 ax-resscn 9003 . . . . . . . . . . . . 13  |-  RR  C_  CC
17 ssid 3327 . . . . . . . . . . . . . . . 16  |-  CC  C_  CC
18 cncfss 18882 . . . . . . . . . . . . . . . 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 3306 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
2114negfcncf 18902 . . . . . . . . . . . . . 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 18881 . . . . . . . . . . . . 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 9037 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2827prid1 3872 . . . . . . . . . . . . . 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 5829 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  RR )
3231recnd 9070 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  CC )
33 fvex 5701 . . . . . . . . . . . . . 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 5738 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  =  ( w  e.  ( A (,) B
)  |->  ( F `  w ) ) )
3635oveq2d 6056 . . . . . . . . . . . . . 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 19790 . . . . . . . . . . . . . . . . . 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 5540 . . . . . . . . . . . . . . . . 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 5738 . . . . . . . . . . . . . 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 2438 . . . . . . . . . . . . 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 19805 . . . . . . . . . . . 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 5031 . . . . . . . . . . 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 5326 . . . . . . . . . . . 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 9260 . . . . . . . . . . . . 13  |-  -u (
( RR  _D  F
) `  w )  e.  _V
4948a1i 11 . . . . . . . . . . . 12  |-  ( w  e.  ( A (,) B )  ->  -u (
( RR  _D  F
) `  w )  e.  _V )
5047, 49mprg 2735 . . . . . . . . . . 11  |-  dom  (
w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) )  =  ( A (,) B )
5146, 50syl6eq 2452 . . . . . . . . . 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 5830 . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  dom  ( RR  _D  F ) )
5738, 56ffvelrnd 5830 . . . . . . . . . . . . . 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 10948 . . . . . . . . . . . . . . . 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 3309 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  RR )
63 iccneg 10974 . . . . . . . . . . . . 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 5689 . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . 16  |-  ( w  =  N  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  N ) )
6867negeqd 9256 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  N ) )
69 eqid 2404 . . . . . . . . . . . . . . 15  |-  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)
70 negex 9260 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  N )  e.  _V
7168, 69, 70fvmpt 5765 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . 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 5689 . . . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . 16  |-  ( w  =  M  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  M ) )
7675negeqd 9256 . . . . . . . . . . . . . . 15  |-  ( w  =  M  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  M ) )
77 negex 9260 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  M )  e.  _V
7876, 69, 77fvmpt 5765 . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . 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 6058 . . . . . . . . . . 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 2481 . . . . . . . . . 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 2404 . . . . . . . . . 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 19846 . . . . . . . . 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 5056 . . . . . . . . 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 2480 . . . . . . . 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 9260 . . . . . . . . 9  |-  -u x  e.  _V
8869elrnmpt 5076 . . . . . . . . 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 9070 . . . . . . . . . . 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 19798 . . . . . . . . . 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 9363 . . . . . . . . 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 2406 . . . . . . . . 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 2683 . . . . . . 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 5550 . . . . . . . 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 5733 . . . . . . 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 3314 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
106 fveq2 5687 . . . . . 6  |-  ( M  =  N  ->  (
( RR  _D  F
) `  M )  =  ( ( RR 
_D  F ) `  N ) )
107106oveq1d 6055 . . . . 5  |-  ( M  =  N  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  ( ( ( RR  _D  F ) `
 N ) [,] ( ( RR  _D  F ) `  N
) ) )
10857rexrd 9090 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR* )
109 iccid 10917 . . . . . 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 2458 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  { ( ( RR  _D  F ) `
 N ) } )
112 ffn 5550 . . . . . . . 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 5826 . . . . . . 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 3903 . . . . 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 3342 . . 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 2404 . . . . . 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 19846 . . . . 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 3314 . . 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 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916    C_ wss 3280   {csn 3774   {cpr 3775   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945    x. cmul 8951   RR*cxr 9075    < clt 9076    - cmin 9247   -ucneg 9248   (,)cioo 10872   [,]cicc 10875   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  dvne0  19848
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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