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Theorem isder 25106
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
StepHypRef Expression
1 isder.5 . . 3  |-  D  =  ( N der I
)
21oveqi 5832 . 2  |-  ( F D P )  =  ( F ( N der I ) P )
3 simpl1 963 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  N  e.  NN )
4 simpl2 964 . . . . 5  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  I  e.  Intvl )
5 ovex 5844 . . . . . 6  |-  ( ( RR  ^m  ( 1 ... N ) )  ^m  I )  e. 
_V
6 mpt2exga 6158 . . . . . 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 647 . . . . 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 445 . . . . . . . . . 10  |-  ( ( n  =  N  /\  i  =  I )  ->  n  =  N )
98oveq2d 5835 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( 1 ... n
)  =  ( 1 ... N ) )
109oveq2d 5835 . . . . . . . 8  |-  ( ( n  =  N  /\  i  =  I )  ->  ( RR  ^m  (
1 ... n ) )  =  ( RR  ^m  ( 1 ... N
) ) )
11 simpr 449 . . . . . . . 8  |-  ( ( n  =  N  /\  i  =  I )  ->  i  =  I )
1210, 11oveq12d 5837 . . . . . . 7  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( RR  ^m  ( 1 ... n
) )  ^m  i
)  =  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) )
13 mpteq1 4101 . . . . . . . . . . . 12  |-  ( ( 1 ... n )  =  ( 1 ... N )  ->  (
x  e.  ( 1 ... n )  |->  (
topGen `  ran  (,) )
)  =  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) )
149, 13syl 17 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( x  e.  ( 1 ... n ) 
|->  ( topGen `  ran  (,) )
)  =  ( x  e.  ( 1 ... N )  |->  ( topGen ` 
ran  (,) ) ) )
1514fveq2d 5489 . . . . . . . . . 10  |-  ( ( n  =  N  /\  i  =  I )  ->  (  topX  `  ( x  e.  ( 1 ... n )  |->  ( topGen ` 
ran  (,) ) ) )  =  (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) ) )
1615oveq1d 5834 . . . . . . . . 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 3294 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( i  \  {
p } )  =  ( I  \  {
p } ) )
1816, 17fveq12d 5491 . . . . . . . 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 5489 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( / cv `  n
)  =  ( / cv `  N ) )
208fveq2d 5489 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  i  =  I )  ->  (  - cv  `  n
)  =  (  - cv  `  N ) )
2120oveqd 5836 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( f `  x ) (  - cv  `  n ) ( f `  p ) )  =  ( ( f `  x ) (  - cv  `  N
) ( f `  p ) ) )
22 eqidd 2285 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( x  -  p
)  =  ( x  -  p ) )
2319, 21, 22oveq123d 5840 . . . . . . . . . 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 4099 . . . . . . . . 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 3804 . . . . . . . 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 5491 . . . . . . 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 5871 . . . . . 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 25104 . . . . . 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 5938 . . . . 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 1187 . . . 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 5836 . . 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 5844 . . . . . . . . 9  |-  ( RR 
^m  ( 1 ... N ) )  e. 
_V
33 simpr2 967 . . . . . . . . 9  |-  ( ( I  =/=  { P }  /\  ( N  e.  NN  /\  I  e. 
Intvl  /\  P  e.  I
) )  ->  I  e.  Intvl )
34 elmapg 6780 . . . . . . . . 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 647 . . . . . . . 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 608 . . . . . . 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 74 . . . . . 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 420 . . . . 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 421 . . . 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 965 . . . 4  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  P  e.  I )
41 fvex 5499 . . . . 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 12 . . . 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 3652 . . . . . . . . 9  |-  ( p  =  P  ->  { p }  =  { P } )
4443adantl 454 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  { p }  =  { P } )
4544difeq2d 3295 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( I  \  {
p } )  =  ( I  \  { P } ) )
4645fveq2d 5489 . . . . . 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 449 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  p  =  P )
48 simpl 445 . . . . . . . . . . 11  |-  ( ( f  =  F  /\  p  =  P )  ->  f  =  F )
4948fveq1d 5487 . . . . . . . . . 10  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f `  x
)  =  ( F `
 x ) )
5048, 47fveq12d 5491 . . . . . . . . . 10  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f `  p
)  =  ( F `
 P ) )
5149, 50oveq12d 5837 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f `  x ) (  - cv  `  N ) ( f `  p ) )  =  ( ( F `  x ) (  - cv  `  N
) ( F `  P ) ) )
5247oveq2d 5835 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( x  -  p
)  =  ( x  -  P ) )
5351, 52oveq12d 5837 . . . . . . . 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 4099 . . . . . . 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 3805 . . . . . 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 5491 . . . . 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 2284 . . . . 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 5938 . . . 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 1187 . . 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 2288 . . . . . . 7  |-  (  topX  `  ( x  e.  ( 1 ... N ) 
|->  ( topGen `  ran  (,) )
) )  =  J
6261a1i 12 . . . . . 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 2288 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  K
6564a1i 12 . . . . . 6  |-  ( ( ( N  e.  NN  /\  I  e.  Intvl  /\  P  e.  I )  /\  (
I  =/=  { P }  /\  F : I --> ( RR  ^m  (
1 ... N ) ) ) )  ->  ( topGen `
 ran  (,) )  =  K )
6662, 65oveq12d 5837 . . . . 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 5487 . . . 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 2288 . . . . . 6  |-  ( x  e.  ( I  \  { P } )  |->  ( ( ( F `  x ) (  - cv  `  N ) ( F `  P ) ) ( / cv `  N ) ( x  -  P ) ) )  =  S
7069a1i 12 . . . . 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 3804 . . . 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 5491 . . 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 2320 . 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 2328 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 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    =/= wne 2447   _Vcvv 2789    \ cdif 3150   {csn 3641   <.cop 3644    e. cmpt 4078   ran crn 4689   -->wf 5217   ` cfv 5221  (class class class)co 5819    e. cmpt2 5821    ^m cmap 6767   RRcr 8731   1c1 8733    - cmin 9032   NNcn 9741   (,)cioo 10650   ...cfz 10776   topGenctg 13336    topX ctopx 24943    fLimfrs cflimfrs 24977    - cv cmcv 25063   / cvcdivcv 25090   Intvlcintvl 25095   dercder 25103
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-1st 6083  df-2nd 6084  df-map 6769  df-der 25104
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