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Theorem isder 25039
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 5770 . 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 5782 . . . . . 6  |-  ( ( RR  ^m  ( 1 ... N ) )  ^m  I )  e. 
_V
6 mpt2exga 6096 . . . . . 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 5773 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( 1 ... n
)  =  ( 1 ... N ) )
109oveq2d 5773 . . . . . . . 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 5775 . . . . . . 7  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( RR  ^m  ( 1 ... n
) )  ^m  i
)  =  ( ( RR  ^m  ( 1 ... N ) )  ^m  I ) )
13 mpteq1 4040 . . . . . . . . . . . 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 5427 . . . . . . . . . 10  |-  ( ( n  =  N  /\  i  =  I )  ->  (  topX  `  ( x  e.  ( 1 ... n )  |->  ( topGen ` 
ran  (,) ) ) )  =  (  topX  `  (
x  e.  ( 1 ... N )  |->  (
topGen `  ran  (,) )
) ) )
1615oveq1d 5772 . . . . . . . . 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 3235 . . . . . . . . 9  |-  ( ( n  =  N  /\  i  =  I )  ->  ( i  \  {
p } )  =  ( I  \  {
p } ) )
1816, 17fveq12d 5429 . . . . . . . 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 5427 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( / cv `  n
)  =  ( / cv `  N ) )
208fveq2d 5427 . . . . . . . . . . . 12  |-  ( ( n  =  N  /\  i  =  I )  ->  (  - cv  `  n
)  =  (  - cv  `  N ) )
2120oveqd 5774 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( ( f `  x ) (  - cv  `  n ) ( f `  p ) )  =  ( ( f `  x ) (  - cv  `  N
) ( f `  p ) ) )
22 eqidd 2257 . . . . . . . . . . 11  |-  ( ( n  =  N  /\  i  =  I )  ->  ( x  -  p
)  =  ( x  -  p ) )
2319, 21, 22oveq123d 5778 . . . . . . . . . 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 4038 . . . . . . . . 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 3744 . . . . . . . 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 5429 . . . . . . 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 5809 . . . . . 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 25037 . . . . . 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 5876 . . . . 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 5774 . . 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 5782 . . . . . . . . 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 6718 . . . . . . . . 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 5437 . . . . 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 3592 . . . . . . . . 9  |-  ( p  =  P  ->  { p }  =  { P } )
4443adantl 454 . . . . . . . 8  |-  ( ( f  =  F  /\  p  =  P )  ->  { p }  =  { P } )
4544difeq2d 3236 . . . . . . 7  |-  ( ( f  =  F  /\  p  =  P )  ->  ( I  \  {
p } )  =  ( I  \  { P } ) )
4645fveq2d 5427 . . . . . 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 5425 . . . . . . . . . 10  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f `  x
)  =  ( F `
 x ) )
5048, 47fveq12d 5429 . . . . . . . . . 10  |-  ( ( f  =  F  /\  p  =  P )  ->  ( f `  p
)  =  ( F `
 P ) )
5149, 50oveq12d 5775 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( ( f `  x ) (  - cv  `  N ) ( f `  p ) )  =  ( ( F `  x ) (  - cv  `  N
) ( F `  P ) ) )
5247oveq2d 5773 . . . . . . . . 9  |-  ( ( f  =  F  /\  p  =  P )  ->  ( x  -  p
)  =  ( x  -  P ) )
5351, 52oveq12d 5775 . . . . . . . 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 4038 . . . . . . 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 3745 . . . . . 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 5429 . . . . 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 2256 . . . . 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 5876 . . . 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 2260 . . . . . . 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 2260 . . . . . . 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 5775 . . . . 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 5425 . . . 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 2260 . . . . . 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 3744 . . . 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 5429 . . 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 2292 . 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 2300 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 1619    e. wcel 1621    =/= wne 2419   _Vcvv 2740    \ cdif 3091   {csn 3581   <.cop 3584    e. cmpt 4017   ran crn 4627   -->wf 4634   ` cfv 4638  (class class class)co 5757    e. cmpt2 5759    ^m cmap 6705   RRcr 8669   1c1 8671    - cmin 8970   NNcn 9679   (,)cioo 10587   ...cfz 10713   topGenctg 13269    topX ctopx 24876    fLimfrs cflimfrs 24910    - cv cmcv 24996   / cvcdivcv 25023   Intvlcintvl 25028   dercder 25036
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-map 6707  df-der 25037
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