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Theorem dvivth 23041
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 22487 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 dvivth.1 . . . . . . . . . 10  |-  ( ph  ->  M  e.  ( A (,) B ) )
21adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  e.  ( A (,) B ) )
3 dvivth.2 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ( A (,) B ) )
43adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  e.  ( A (,) B ) )
5 dvivth.3 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> RR ) )
6 cncff 22003 . . . . . . . . . . . . . . 15  |-  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  F :
( A (,) B
) --> RR )
75, 6syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  F : ( A (,) B ) --> RR )
87ffvelrnda 6037 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  ( F `  w )  e.  RR )
98renegcld 10067 . . . . . . . . . . . 12  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  -u ( F `
 w )  e.  RR )
10 eqid 2471 . . . . . . . . . . . 12  |-  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)
119, 10fmptd 6061 . . . . . . . . . . 11  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR )
12 ax-resscn 9614 . . . . . . . . . . . 12  |-  RR  C_  CC
13 ssid 3437 . . . . . . . . . . . . . . 15  |-  CC  C_  CC
14 cncfss 22009 . . . . . . . . . . . . . . 15  |-  ( ( RR  C_  CC  /\  CC  C_  CC )  ->  (
( A (,) B
) -cn-> RR )  C_  (
( A (,) B
) -cn-> CC ) )
1512, 13, 14mp2an 686 . . . . . . . . . . . . . 14  |-  ( ( A (,) B )
-cn-> RR )  C_  (
( A (,) B
) -cn-> CC )
1615, 5sseldi 3416 . . . . . . . . . . . . 13  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
1710negfcncf 22029 . . . . . . . . . . . . 13  |-  ( F  e.  ( ( A (,) B ) -cn-> CC )  ->  ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> CC ) )
1816, 17syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> CC ) )
19 cncffvrn 22008 . . . . . . . . . . . 12  |-  ( ( 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 ) )
2012, 18, 19sylancr 676 . . . . . . . . . . 11  |-  ( ph  ->  ( ( w  e.  ( A (,) B
)  |->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR )  <-> 
( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR ) )
2111, 20mpbird 240 . . . . . . . . . 10  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) )  e.  ( ( A (,) B ) -cn-> RR ) )
2221adantr 472 . . . . . . . . 9  |-  ( (
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 ) )
23 reelprrecn 9649 . . . . . . . . . . . . 13  |-  RR  e.  { RR ,  CC }
2423a1i 11 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  RR  e.  { RR ,  CC } )
257adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F : ( A (,) B ) --> RR )
2625ffvelrnda 6037 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  RR )
2726recnd 9687 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  CC )
28 fvex 5889 . . . . . . . . . . . . 13  |-  ( ( RR  _D  F ) `
 w )  e. 
_V
2928a1i 11 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( ( RR  _D  F ) `  w
)  e.  _V )
3025feqmptd 5932 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  =  ( w  e.  ( A (,) B
)  |->  ( F `  w ) ) )
3130oveq2d 6324 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
)  =  ( RR 
_D  ( w  e.  ( A (,) B
)  |->  ( F `  w ) ) ) )
32 ioossre 11721 . . . . . . . . . . . . . . . . 17  |-  ( A (,) B )  C_  RR
33 dvfre 22984 . . . . . . . . . . . . . . . . 17  |-  ( ( F : ( A (,) B ) --> RR 
/\  ( A (,) B )  C_  RR )  ->  ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR )
347, 32, 33sylancl 675 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( RR  _D  F
) : dom  ( RR  _D  F ) --> RR )
35 dvivth.4 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
3635feq2d 5725 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> RR  <->  ( RR  _D  F ) : ( A (,) B ) --> RR ) )
3734, 36mpbid 215 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> RR )
3837adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
) : ( A (,) B ) --> RR )
3938feqmptd 5932 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
)  =  ( w  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 w ) ) )
4031, 39eqtr3d 2507 . . . . . . . . . . . 12  |-  ( (
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 )
) )
4124, 27, 29, 40dvmptneg 22999 . . . . . . . . . . 11  |-  ( (
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
) ) )
4241dmeqd 5042 . . . . . . . . . 10  |-  ( (
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
) ) )
43 dmmptg 5339 . . . . . . . . . . 11  |-  ( A. w  e.  ( A (,) B ) -u (
( RR  _D  F
) `  w )  e.  _V  ->  dom  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  =  ( A (,) B ) )
44 negex 9893 . . . . . . . . . . . 12  |-  -u (
( RR  _D  F
) `  w )  e.  _V
4544a1i 11 . . . . . . . . . . 11  |-  ( w  e.  ( A (,) B )  ->  -u (
( RR  _D  F
) `  w )  e.  _V )
4643, 45mprg 2770 . . . . . . . . . 10  |-  dom  (
w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) )  =  ( A (,) B )
4742, 46syl6eq 2521 . . . . . . . . 9  |-  ( (
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 ) )
48 simprl 772 . . . . . . . . 9  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  <  N )
49 simprr 774 . . . . . . . . . . 11  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) ) )
5037, 1ffvelrnd 6038 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( RR  _D  F ) `  M
)  e.  RR )
5150adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  F ) `  M
)  e.  RR )
523, 35eleqtrrd 2552 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  dom  ( RR  _D  F ) )
5334, 52ffvelrnd 6038 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR )
5453adantr 472 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( RR  _D  F ) `  N
)  e.  RR )
55 iccssre 11741 . . . . . . . . . . . . . . 15  |-  ( ( ( ( RR  _D  F ) `  M
)  e.  RR  /\  ( ( RR  _D  F ) `  N
)  e.  RR )  ->  ( ( ( RR  _D  F ) `
 M ) [,] ( ( RR  _D  F ) `  N
) )  C_  RR )
5650, 53, 55syl2anc 673 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  RR )
5756adantr 472 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  RR )
5857, 49sseldd 3419 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  RR )
59 iccneg 11779 . . . . . . . . . . . 12  |-  ( ( ( ( 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 )
) ) )
6051, 54, 58, 59syl3anc 1292 . . . . . . . . . . 11  |-  ( (
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 )
) ) )
6149, 60mpbid 215 . . . . . . . . . 10  |-  ( (
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 )
) )
6241fveq1d 5881 . . . . . . . . . . . 12  |-  ( (
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
) )
63 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  N ) )
6463negeqd 9889 . . . . . . . . . . . . . 14  |-  ( w  =  N  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  N ) )
65 eqid 2471 . . . . . . . . . . . . . 14  |-  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)
66 negex 9893 . . . . . . . . . . . . . 14  |-  -u (
( RR  _D  F
) `  N )  e.  _V
6764, 65, 66fvmpt 5963 . . . . . . . . . . . . 13  |-  ( N  e.  ( A (,) B )  ->  (
( w  e.  ( A (,) B ) 
|->  -u ( ( RR 
_D  F ) `  w ) ) `  N )  =  -u ( ( RR  _D  F ) `  N
) )
684, 67syl 17 . . . . . . . . . . . 12  |-  ( (
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
) )
6962, 68eqtrd 2505 . . . . . . . . . . 11  |-  ( (
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 )
)
7041fveq1d 5881 . . . . . . . . . . . 12  |-  ( (
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
) )
71 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( w  =  M  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  M ) )
7271negeqd 9889 . . . . . . . . . . . . . 14  |-  ( w  =  M  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  M ) )
73 negex 9893 . . . . . . . . . . . . . 14  |-  -u (
( RR  _D  F
) `  M )  e.  _V
7472, 65, 73fvmpt 5963 . . . . . . . . . . . . 13  |-  ( M  e.  ( A (,) B )  ->  (
( w  e.  ( A (,) B ) 
|->  -u ( ( RR 
_D  F ) `  w ) ) `  M )  =  -u ( ( RR  _D  F ) `  M
) )
752, 74syl 17 . . . . . . . . . . . 12  |-  ( (
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
) )
7670, 75eqtrd 2505 . . . . . . . . . . 11  |-  ( (
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 )
)
7769, 76oveq12d 6326 . . . . . . . . . 10  |-  ( (
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 )
) )
7861, 77eleqtrrd 2552 . . . . . . . . 9  |-  ( (
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
) ) )
79 eqid 2471 . . . . . . . . 9  |-  ( 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 ) ) )
802, 4, 22, 47, 48, 78, 79dvivthlem2 23040 . . . . . . . 8  |-  ( (
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 ) ) ) )
8141rneqd 5068 . . . . . . . 8  |-  ( (
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
) ) )
8280, 81eleqtrd 2551 . . . . . . 7  |-  ( (
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
) ) )
83 negex 9893 . . . . . . . 8  |-  -u x  e.  _V
8465elrnmpt 5087 . . . . . . . 8  |-  ( -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
) ) )
8583, 84ax-mp 5 . . . . . . 7  |-  ( -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
) )
8682, 85sylib 201 . . . . . 6  |-  ( (
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 ) )
8758recnd 9687 . . . . . . . . . 10  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  CC )
8887adantr 472 . . . . . . . . 9  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  ->  x  e.  CC )
8924, 27, 29, 40dvmptcl 22992 . . . . . . . . 9  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( ( RR  _D  F ) `  w
)  e.  CC )
9088, 89neg11ad 10001 . . . . . . . 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
)  <->  x  =  (
( RR  _D  F
) `  w )
) )
91 eqcom 2478 . . . . . . . 8  |-  ( x  =  ( ( RR 
_D  F ) `  w )  <->  ( ( RR  _D  F ) `  w )  =  x )
9290, 91syl6bb 269 . . . . . . 7  |-  ( ( ( 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 ) )
9392rexbidva 2889 . . . . . 6  |-  ( (
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 ) )
9486, 93mpbid 215 . . . . 5  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  E. w  e.  ( A (,) B ) ( ( RR  _D  F
) `  w )  =  x )
95 ffn 5739 . . . . . . 7  |-  ( ( RR  _D  F ) : ( A (,) B ) --> RR  ->  ( RR  _D  F )  Fn  ( A (,) B ) )
9638, 95syl 17 . . . . . 6  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  -> 
( RR  _D  F
)  Fn  ( A (,) B ) )
97 fvelrnb 5926 . . . . . 6  |-  ( ( RR  _D  F )  Fn  ( A (,) B )  ->  (
x  e.  ran  ( RR  _D  F )  <->  E. w  e.  ( A (,) B
) ( ( RR 
_D  F ) `  w )  =  x ) )
9896, 97syl 17 . . . . 5  |-  ( (
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 ) )
9994, 98mpbird 240 . . . 4  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ran  ( RR 
_D  F ) )
10099expr 626 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  ->  x  e.  ran  ( RR  _D  F
) ) )
101100ssrdv 3424 . 2  |-  ( (
ph  /\  M  <  N )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
102 fveq2 5879 . . . . 5  |-  ( M  =  N  ->  (
( RR  _D  F
) `  M )  =  ( ( RR 
_D  F ) `  N ) )
103102oveq1d 6323 . . . 4  |-  ( M  =  N  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  ( ( ( RR  _D  F ) `
 N ) [,] ( ( RR  _D  F ) `  N
) ) )
10453rexrd 9708 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR* )
105 iccid 11706 . . . . 5  |-  ( ( ( RR  _D  F
) `  N )  e.  RR*  ->  ( (
( RR  _D  F
) `  N ) [,] ( ( RR  _D  F ) `  N
) )  =  {
( ( RR  _D  F ) `  N
) } )
106104, 105syl 17 . . . 4  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  N ) [,] (
( RR  _D  F
) `  N )
)  =  { ( ( RR  _D  F
) `  N ) } )
107103, 106sylan9eqr 2527 . . 3  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  { ( ( RR  _D  F ) `
 N ) } )
108 ffn 5739 . . . . . . 7  |-  ( ( RR  _D  F ) : dom  ( RR 
_D  F ) --> RR 
->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
10934, 108syl 17 . . . . . 6  |-  ( ph  ->  ( RR  _D  F
)  Fn  dom  ( RR  _D  F ) )
110 fnfvelrn 6034 . . . . . 6  |-  ( ( ( RR  _D  F
)  Fn  dom  ( RR  _D  F )  /\  N  e.  dom  ( RR 
_D  F ) )  ->  ( ( RR 
_D  F ) `  N )  e.  ran  ( RR  _D  F
) )
111109, 52, 110syl2anc 673 . . . . 5  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  ran  ( RR  _D  F ) )
112111snssd 4108 . . . 4  |-  ( ph  ->  { ( ( RR 
_D  F ) `  N ) }  C_  ran  ( RR  _D  F
) )
113112adantr 472 . . 3  |-  ( (
ph  /\  M  =  N )  ->  { ( ( RR  _D  F
) `  N ) }  C_  ran  ( RR 
_D  F ) )
114107, 113eqsstrd 3452 . 2  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) ) 
C_  ran  ( RR  _D  F ) )
1153adantr 472 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  e.  ( A (,) B ) )
1161adantr 472 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  M  e.  ( A (,) B ) )
1175adantr 472 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  e.  ( ( A (,) B ) -cn-> RR ) )
11835adantr 472 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  dom  ( RR  _D  F
)  =  ( A (,) B ) )
119 simprl 772 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  N  <  M )
120 simprr 774 . . . . 5  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) ) )
121 eqid 2471 . . . . 5  |-  ( y  e.  ( A (,) B )  |->  ( ( F `  y )  -  ( x  x.  y ) ) )  =  ( y  e.  ( A (,) B
)  |->  ( ( F `
 y )  -  ( x  x.  y
) ) )
122115, 116, 117, 118, 119, 120, 121dvivthlem2 23040 . . . 4  |-  ( (
ph  /\  ( N  <  M  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  ran  ( RR 
_D  F ) )
123122expr 626 . . 3  |-  ( (
ph  /\  N  <  M )  ->  ( x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  ->  x  e.  ran  ( RR  _D  F
) ) )
124123ssrdv 3424 . 2  |-  ( (
ph  /\  N  <  M )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
12532, 1sseldi 3416 . . 3  |-  ( ph  ->  M  e.  RR )
12632, 3sseldi 3416 . . 3  |-  ( ph  ->  N  e.  RR )
127125, 126lttri4d 9793 . 2  |-  ( ph  ->  ( M  <  N  \/  M  =  N  \/  N  <  M ) )
128101, 114, 124, 127mpjao3dan 1361 1  |-  ( ph  ->  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
)  C_  ran  ( RR 
_D  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    C_ wss 3390   {csn 3959   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556    x. cmul 9562   RR*cxr 9692    < clt 9693    - cmin 9880   -ucneg 9881   (,)cioo 11660   [,]cicc 11663   -cn->ccncf 21986    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by:  dvne0  23042
  Copyright terms: Public domain W3C validator