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Theorem isder 25810
Description: The derivative of  F at point  P is the limit of the slope  F ( x )  -  F ( P )  /  x  -  P when  x tends to  P. Definition 1 of [BourbakiFVR] p. I.11. (Contributed by FL, 29-May-2014.)
Hypotheses
Ref Expression
isder.1  |-  J  =  (  topX  `  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) )
isder.2  |-  K  =  ( topGen `  ran  (,) )
isder.4  |-  S  =  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x
) (  - cv  `  N ) ( F `
 P ) ) ( / cv `  N
) ( x  -  P ) ) )
isder.5  |-  D  =  ( N der I
)
Assertion
Ref Expression
isder  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( F D P )  =  ( ( ( J 
fLimfrs  K ) `  (
I  \  { P } ) ) `  <. P ,  S >. ) )
Distinct variable groups:    x, I    x, F    x, N    x, P
Allowed substitution hints:    D( x)    S( x)    J( x)    K( x)

Proof of Theorem isder
Dummy variables  f 
i  n  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isder.5 . . 3  |-  D  =  ( N der I
)
21oveqi 5887 . 2  |-  ( F D P )  =  ( F ( N der I ) P )
3 simpl1 958 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  N  e.  NN )
4 simpl2 959 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  I  e.  Intvl )
5 ovex 5899 . . . . . 6  |-  ( ( RR  ^m  ( 1 ... N ) )  ^m  I )  e. 
_V
6 mpt2exga 6213 . . . . . 6  |-  ( ( ( ( RR  ^m  ( 1 ... N
) )  ^m  I
)  e.  _V  /\  I  e.  Intvl )  -> 
( f  e.  ( ( RR  ^m  (
1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) )  e.  _V )
75, 4, 6sylancr 644 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  (
f  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) )  e.  _V )
8 simpl 443 . . . . . . . . . 10  |-  ( ( n  =  N  /\  i  =  I )  ->  n  =  N )
98oveq2d 5890 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( 1 ... n
)  =  ( 1 ... N ) )
109oveq2d 5890 . . . . . . . 8  |-  ( ( n  =  N  /\  i  =  I )  ->  ( RR  ^m  (
1 ... n ) )  =  ( RR  ^m  ( 1 ... N
) ) )
11 simpr 447 . . . . . . . 8  |-  ( ( n  =  N  /\  i  =  I )  ->  i  =  I )
1210, 11oveq12d 5892 . . . . . . 7  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( RR  ^m  ( 1 ... n
) )  ^m  i
)  =  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) )
13 mpteq1 4116 . . . . . . . . . . . 12  |-  ( ( 1 ... n )  =  ( 1 ... N )  ->  (
x  e.  ( 1 ... n )  |->  (
topGen `  ran  (,) )
)  =  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) )
149, 13syl 15 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( x  e.  ( 1 ... n ) 
|->  ( topGen `  ran  (,) )
)  =  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) )
1514fveq2d 5545 . . . . . . . . . 10  |-  ( ( n  =  N  /\  i  =  I )  ->  (  topX  `  ( x  e.  ( 1 ... n )  |->  ( topGen ` 
ran  (,) ) ) )  =  (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) ) )
1615oveq1d 5889 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( (  topX  `  (
x  e.  ( 1 ... n )  |->  (
topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) )  =  ( (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) )
1711difeq1d 3306 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( i  \  {
p } )  =  ( I  \  {
p } ) )
1816, 17fveq12d 5547 . . . . . . . 8  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( (  topX  `  ( x  e.  ( 1 ... n ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( i  \  {
p } ) )  =  ( ( ( 
topX  `  ( x  e.  ( 1 ... N
)  |->  ( topGen `  ran  (,) ) ) )  fLimfrs  (
topGen `  ran  (,) )
) `  ( I  \  { p } ) ) )
198fveq2d 5545 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( / cv `  n
)  =  ( / cv `  N ) )
208fveq2d 5545 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  i  =  I )  ->  (  - cv  `  n
)  =  (  - cv  `  N ) )
2120oveqd 5891 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( f `  x ) (  - cv  `  n ) ( f `  p ) )  =  ( ( f `  x ) (  - cv  `  N
) ( f `  p ) ) )
22 eqidd 2297 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( x  -  p
)  =  ( x  -  p ) )
2319, 21, 22oveq123d 5895 . . . . . . . . . 10  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( ( f `
 x ) (  - cv  `  n
) ( f `  p ) ) ( / cv `  n
) ( x  -  p ) )  =  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
2417, 23mpteq12dv 4114 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( x  e.  ( i  \  { p } )  |->  ( ( ( f `  x
) (  - cv  `  n ) ( f `
 p ) ) ( / cv `  n
) ( x  -  p ) ) )  =  ( x  e.  ( I  \  {
p } )  |->  ( ( ( f `  x ) (  - cv  `  N ) ( f `  p ) ) ( / cv `  N ) ( x  -  p ) ) ) )
2524opeq2d 3819 . . . . . . . 8  |-  ( ( n  =  N  /\  i  =  I )  -> 
<. p ,  ( x  e.  ( i  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  n
) ( f `  p ) ) ( / cv `  n
) ( x  -  p ) ) )
>.  =  <. p ,  ( x  e.  ( I  \  { p } )  |->  ( ( ( f `  x
) (  - cv  `  N ) ( f `
 p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. )
2618, 25fveq12d 5547 . . . . . . 7  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( ( ( 
topX  `  ( x  e.  ( 1 ... n
)  |->  ( topGen `  ran  (,) ) ) )  fLimfrs  (
topGen `  ran  (,) )
) `  ( i  \  { p } ) ) `  <. p ,  ( x  e.  ( i  \  {
p } )  |->  ( ( ( f `  x ) (  - cv  `  n ) ( f `  p ) ) ( / cv `  n ) ( x  -  p ) ) ) >. )  =  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) )
2712, 11, 26mpt2eq123dv 5926 . . . . . 6  |-  ( ( n  =  N  /\  i  =  I )  ->  ( f  e.  ( ( RR  ^m  (
1 ... n ) )  ^m  i ) ,  p  e.  i  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... n ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( i  \  {
p } ) ) `
 <. p ,  ( x  e.  ( i 
\  { p }
)  |->  ( ( ( f `  x ) (  - cv  `  n
) ( f `  p ) ) ( / cv `  n
) ( x  -  p ) ) )
>. ) )  =  ( f  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) ) )
28 df-der 25808 . . . . . 6  |-  der  =  ( n  e.  NN ,  i  e.  Intvl  |->  ( f  e.  ( ( RR 
^m  ( 1 ... n ) )  ^m  i ) ,  p  e.  i  |->  ( ( ( (  topX  `  (
x  e.  ( 1 ... n )  |->  (
topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( i  \  {
p } ) ) `
 <. p ,  ( x  e.  ( i 
\  { p }
)  |->  ( ( ( f `  x ) (  - cv  `  n
) ( f `  p ) ) ( / cv `  n
) ( x  -  p ) ) )
>. ) ) )
2927, 28ovmpt2ga 5993 . . . . 5  |-  ( ( N  e.  NN  /\  I  e.  Intvl  /\  (
f  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) )  e.  _V )  ->  ( N der I )  =  ( f  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) ) )
303, 4, 7, 29syl3anc 1182 . . . 4  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( N der I )  =  ( f  e.  ( ( RR  ^m  (
1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) ) )
3130oveqd 5891 . . 3  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( F ( N der I ) P )  =  ( F ( f  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) ) P ) )
32 ovex 5899 . . . . . . . . 9  |-  ( RR 
^m  ( 1 ... N ) )  e. 
_V
33 simpr2 962 . . . . . . . . 9  |-  ( ( I  =/=  { P }  /\  ( N  e.  NN  /\  I  e. 
Intvl  /\  P  e.  I
) )  ->  I  e.  Intvl )
34 elmapg 6801 . . . . . . . . 9  |-  ( ( ( RR  ^m  (
1 ... N ) )  e.  _V  /\  I  e.  Intvl )  ->  ( F  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I )  <->  F :
I --> ( RR  ^m  ( 1 ... N
) ) ) )
3532, 33, 34sylancr 644 . . . . . . . 8  |-  ( ( I  =/=  { P }  /\  ( N  e.  NN  /\  I  e. 
Intvl  /\  P  e.  I
) )  ->  ( F  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I )  <->  F :
I --> ( RR  ^m  ( 1 ... N
) ) ) )
3635exbiri 605 . . . . . . 7  |-  ( I  =/=  { P }  ->  ( ( N  e.  NN  /\  I  e. 
Intvl  /\  P  e.  I
)  ->  ( F : I --> ( RR 
^m  ( 1 ... N ) )  ->  F  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ) ) )
3736com23 72 . . . . . 6  |-  ( I  =/=  { P }  ->  ( F : I --> ( RR  ^m  (
1 ... N ) )  ->  ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  ->  F  e.  ( ( RR  ^m  ( 1 ... N
) )  ^m  I
) ) ) )
3837imp 418 . . . . 5  |-  ( ( I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) )  ->  ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  ->  F  e.  ( ( RR  ^m  ( 1 ... N
) )  ^m  I
) ) )
3938impcom 419 . . . 4  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  F  e.  ( ( RR  ^m  ( 1 ... N
) )  ^m  I
) )
40 simpl3 960 . . . 4  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  P  e.  I )
41 fvex 5555 . . . . 5  |-  ( ( ( (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { P } ) ) `  <. P ,  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) ) >. )  e.  _V
4241a1i 10 . . . 4  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  (
( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { P } ) ) `  <. P ,  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) ) >. )  e.  _V )
43 sneq 3664 . . . . . . . . 9  |-  ( p  =  P  ->  { p }  =  { P } )
4443adantl 452 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  { p }  =  { P } )
4544difeq2d 3307 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( I  \  {
p } )  =  ( I  \  { P } ) )
4645fveq2d 5545 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) )  =  ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { P } ) ) )
47 simpr 447 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
48 simpl 443 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  p  =  P )  ->  f  =  F )
4948fveq1d 5543 . . . . . . . . . 10  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f `  x
)  =  ( F `
 x ) )
5048, 47fveq12d 5547 . . . . . . . . . 10  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f `  p
)  =  ( F `
 P ) )
5149, 50oveq12d 5892 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f `  x ) (  - cv  `  N ) ( f `  p ) )  =  ( ( F `  x ) (  - cv  `  N
) ( F `  P ) ) )
5247oveq2d 5890 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( x  -  p
)  =  ( x  -  P ) )
5351, 52oveq12d 5892 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) )  =  ( ( ( F `
 x ) (  - cv  `  N
) ( F `  P ) ) ( / cv `  N
) ( x  -  P ) ) )
5445, 53mpteq12dv 4114 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( x  e.  ( I  \  { p } )  |->  ( ( ( f `  x
) (  - cv  `  N ) ( f `
 p ) ) ( / cv `  N
) ( x  -  p ) ) )  =  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x
) (  - cv  `  N ) ( F `
 P ) ) ( / cv `  N
) ( x  -  P ) ) ) )
5547, 54opeq12d 3820 . . . . . 6  |-  ( ( f  =  F  /\  p  =  P )  -> 
<. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>.  =  <. P , 
( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x
) (  - cv  `  N ) ( F `
 P ) ) ( / cv `  N
) ( x  -  P ) ) )
>. )
5646, 55fveq12d 5547 . . . . 5  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( ( ( 
topX  `  ( x  e.  ( 1 ... N
)  |->  ( topGen `  ran  (,) ) ) )  fLimfrs  (
topGen `  ran  (,) )
) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  {
p } )  |->  ( ( ( f `  x ) (  - cv  `  N ) ( f `  p ) ) ( / cv `  N ) ( x  -  p ) ) ) >. )  =  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { P } ) ) `  <. P ,  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) ) >. ) )
57 eqid 2296 . . . . 5  |-  ( f  e.  ( ( RR 
^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) )  =  ( f  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) )
5856, 57ovmpt2ga 5993 . . . 4  |-  ( ( F  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I )  /\  P  e.  I  /\  ( ( ( ( 
topX  `  ( x  e.  ( 1 ... N
)  |->  ( topGen `  ran  (,) ) ) )  fLimfrs  (
topGen `  ran  (,) )
) `  ( I  \  { P } ) ) `  <. P , 
( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x
) (  - cv  `  N ) ( F `
 P ) ) ( / cv `  N
) ( x  -  P ) ) )
>. )  e.  _V )  ->  ( F ( f  e.  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  { p } ) 
|->  ( ( ( f `
 x ) (  - cv  `  N
) ( f `  p ) ) ( / cv `  N
) ( x  -  p ) ) )
>. ) ) P )  =  ( ( ( (  topX  `  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) ) 
fLimfrs  ( topGen `  ran  (,) )
) `  ( I  \  { P } ) ) `  <. P , 
( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x
) (  - cv  `  N ) ( F `
 P ) ) ( / cv `  N
) ( x  -  P ) ) )
>. ) )
5939, 40, 42, 58syl3anc 1182 . . 3  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( F ( f  e.  ( ( RR  ^m  ( 1 ... N
) )  ^m  I
) ,  p  e.  I  |->  ( ( ( (  topX  `  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) ) 
fLimfrs  ( topGen `  ran  (,) )
) `  ( I  \  { p } ) ) `  <. p ,  ( x  e.  ( I  \  {
p } )  |->  ( ( ( f `  x ) (  - cv  `  N ) ( f `  p ) ) ( / cv `  N ) ( x  -  p ) ) ) >. ) ) P )  =  ( ( ( (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { P } ) ) `  <. P ,  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) ) >. ) )
60 isder.1 . . . . . . . 8  |-  J  =  (  topX  `  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) )
6160eqcomi 2300 . . . . . . 7  |-  (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  =  J
6261a1i 10 . . . . . 6  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  (  topX  `  ( x  e.  ( 1 ... N
)  |->  ( topGen `  ran  (,) ) ) )  =  J )
63 isder.2 . . . . . . . 8  |-  K  =  ( topGen `  ran  (,) )
6463eqcomi 2300 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  K
6564a1i 10 . . . . . 6  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( topGen `
 ran  (,) )  =  K )
6662, 65oveq12d 5892 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  (
(  topX  `  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) ) 
fLimfrs  ( topGen `  ran  (,) )
)  =  ( J 
fLimfrs  K ) )
6766fveq1d 5543 . . . 4  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  (
( (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { P } ) )  =  ( ( J  fLimfrs  K ) `  ( I 
\  { P }
) ) )
68 isder.4 . . . . . . 7  |-  S  =  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x
) (  - cv  `  N ) ( F `
 P ) ) ( / cv `  N
) ( x  -  P ) ) )
6968eqcomi 2300 . . . . . 6  |-  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) )  =  S
7069a1i 10 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  (
x  e.  ( I 
\  { P }
)  |->  ( ( ( F `  x ) (  - cv  `  N
) ( F `  P ) ) ( / cv `  N
) ( x  -  P ) ) )  =  S )
7170opeq2d 3819 . . . 4  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  <. P , 
( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x
) (  - cv  `  N ) ( F `
 P ) ) ( / cv `  N
) ( x  -  P ) ) )
>.  =  <. P ,  S >. )
7267, 71fveq12d 5547 . . 3  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  (
( ( (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  fLimfrs  ( topGen ` 
ran  (,) ) ) `  ( I  \  { P } ) ) `  <. P ,  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) ) >. )  =  ( ( ( J  fLimfrs  K ) `  ( I 
\  { P }
) ) `  <. P ,  S >. )
)
7331, 59, 723eqtrd 2332 . 2  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( F ( N der I ) P )  =  ( ( ( J  fLimfrs  K ) `  ( I  \  { P } ) ) `  <. P ,  S >. ) )
742, 73syl5eq 2340 1  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( F D P )  =  ( ( ( J 
fLimfrs  K ) `  (
I  \  { P } ) ) `  <. P ,  S >. ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    \ cdif 3162   {csn 3653   <.cop 3656    e. cmpt 4093   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876    ^m cmap 6788   RRcr 8752   1c1 8754    - cmin 9053   NNcn 9762   (,)cioo 10672   ...cfz 10798   topGenctg 13358    topX ctopx 25647    fLimfrs cflimfrs 25681    - cv cmcv 25767   / cvcdivcv 25794   Intvlcintvl 25799   dercder 25807
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2561  df-rex 2562  df-reu 2563  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-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  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-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-der 25808
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