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Theorem dvivth 19373
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 18834 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 10728 . . . 4  |-  ( A (,) B )  C_  RR
2 dvivth.1 . . . 4  |-  ( ph  ->  M  e.  ( A (,) B ) )
31, 2sseldi 3191 . . 3  |-  ( ph  ->  M  e.  RR )
4 dvivth.2 . . . 4  |-  ( ph  ->  N  e.  ( A (,) B ) )
51, 4sseldi 3191 . . 3  |-  ( ph  ->  N  e.  RR )
63, 5lttri4d 8976 . 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 18413 . . . . . . . . . . . . . . . 16  |-  ( F  e.  ( ( A (,) B ) -cn-> RR )  ->  F :
( A (,) B
) --> RR )
119, 10syl 15 . . . . . . . . . . . . . . 15  |-  ( ph  ->  F : ( A (,) B ) --> RR )
12 ffvelrn 5679 . . . . . . . . . . . . . . 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 9226 . . . . . . . . . . . . 13  |-  ( (
ph  /\  w  e.  ( A (,) B ) )  ->  -u ( F `
 w )  e.  RR )
15 eqid 2296 . . . . . . . . . . . . 13  |-  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u ( F `  w )
)
1614, 15fmptd 5700 . . . . . . . . . . . 12  |-  ( ph  ->  ( w  e.  ( A (,) B ) 
|->  -u ( F `  w ) ) : ( A (,) B
) --> RR )
17 ax-resscn 8810 . . . . . . . . . . . . 13  |-  RR  C_  CC
18 ssid 3210 . . . . . . . . . . . . . . . 16  |-  CC  C_  CC
19 cncfss 18419 . . . . . . . . . . . . . . . 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 3191 . . . . . . . . . . . . . 14  |-  ( ph  ->  F  e.  ( ( A (,) B )
-cn-> CC ) )
2215negfcncf 18438 . . . . . . . . . . . . . 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 18418 . . . . . . . . . . . . 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 8844 . . . . . . . . . . . . . . 15  |-  RR  e.  _V
2928prid1 3747 . . . . . . . . . . . . . 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 8877 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  /\  w  e.  ( A (,) B ) )  -> 
( F `  w
)  e.  CC )
34 fvex 5555 . . . . . . . . . . . . . 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 5591 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  F  =  ( w  e.  ( A (,) B
)  |->  ( F `  w ) ) )
3736oveq2d 5890 . . . . . . . . . . . . . 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 19316 . . . . . . . . . . . . . . . . . 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 5396 . . . . . . . . . . . . . . . . 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 5591 . . . . . . . . . . . . . 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 2330 . . . . . . . . . . . . 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 19331 . . . . . . . . . . . 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 4897 . . . . . . . . . . 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 5186 . . . . . . . . . . . 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 9066 . . . . . . . . . . . . 13  |-  -u (
( RR  _D  F
) `  w )  e.  _V
5049a1i 10 . . . . . . . . . . . 12  |-  ( w  e.  ( A (,) B )  ->  -u (
( RR  _D  F
) `  w )  e.  _V )
5148, 50mprg 2625 . . . . . . . . . . 11  |-  dom  (
w  e.  ( A (,) B )  |->  -u ( ( RR  _D  F ) `  w
) )  =  ( A (,) B )
5247, 51syl6eq 2344 . . . . . . . . . 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 5679 . . . . . . . . . . . . . . 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 2373 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  dom  ( RR  _D  F ) )
59 ffvelrn 5679 . . . . . . . . . . . . . . 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 10747 . . . . . . . . . . . . . . . 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 3194 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( M  <  N  /\  x  e.  ( ( ( RR 
_D  F ) `  M ) [,] (
( RR  _D  F
) `  N )
) ) )  ->  x  e.  RR )
66 iccneg 10773 . . . . . . . . . . . . 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 5543 . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . 16  |-  ( w  =  N  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  N ) )
7170negeqd 9062 . . . . . . . . . . . . . . 15  |-  ( w  =  N  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  N ) )
72 eqid 2296 . . . . . . . . . . . . . . 15  |-  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)  =  ( w  e.  ( A (,) B )  |->  -u (
( RR  _D  F
) `  w )
)
73 negex 9066 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  N )  e.  _V
7471, 72, 73fvmpt 5618 . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . 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 5543 . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . . 16  |-  ( w  =  M  ->  (
( RR  _D  F
) `  w )  =  ( ( RR 
_D  F ) `  M ) )
7978negeqd 9062 . . . . . . . . . . . . . . 15  |-  ( w  =  M  ->  -u (
( RR  _D  F
) `  w )  =  -u ( ( RR 
_D  F ) `  M ) )
80 negex 9066 . . . . . . . . . . . . . . 15  |-  -u (
( RR  _D  F
) `  M )  e.  _V
8179, 72, 80fvmpt 5618 . . . . . . . . . . . . . 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 2328 . . . . . . . . . . . 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 5892 . . . . . . . . . . 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 2373 . . . . . . . . . 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 2296 . . . . . . . . . 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 19372 . . . . . . . . 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 4922 . . . . . . . . 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 2372 . . . . . . . 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 9066 . . . . . . . . 9  |-  -u x  e.  _V
9172elrnmpt 4942 . . . . . . . . 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 8877 . . . . . . . . . . 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 19324 . . . . . . . . . 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 9114 . . . . . . . . . 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 2298 . . . . . . . . 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 2573 . . . . . . 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 5405 . . . . . . . 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 5586 . . . . . . 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 3198 . . 3  |-  ( (
ph  /\  M  <  N )  ->  ( (
( RR  _D  F
) `  M ) [,] ( ( RR  _D  F ) `  N
) )  C_  ran  ( RR  _D  F
) )
110 fveq2 5541 . . . . . 6  |-  ( M  =  N  ->  (
( RR  _D  F
) `  M )  =  ( ( RR 
_D  F ) `  N ) )
111110oveq1d 5889 . . . . 5  |-  ( M  =  N  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  ( ( ( RR  _D  F ) `
 N ) [,] ( ( RR  _D  F ) `  N
) ) )
11260rexrd 8897 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  F ) `  N
)  e.  RR* )
113 iccid 10717 . . . . . 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 2350 . . . 4  |-  ( (
ph  /\  M  =  N )  ->  (
( ( RR  _D  F ) `  M
) [,] ( ( RR  _D  F ) `
 N ) )  =  { ( ( RR  _D  F ) `
 N ) } )
116 ffn 5405 . . . . . . . 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 5678 . . . . . . 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 3776 . . . . 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 3225 . . 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 2296 . . . . . 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 19372 . . . . 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 3198 . . 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 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165   {csn 3653   {cpr 3654   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752    x. cmul 8758   RR*cxr 8882    < clt 8883    - cmin 9053   -ucneg 9054   (,)cioo 10672   [,]cicc 10675   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  dvne0  19374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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