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Theorem cmvth 19828
Description: Cauchy's Mean Value Theorem. If  F ,  G are real continuous functions on  [ A ,  B ] differentiable on  ( A ,  B ), then there is some  x  e.  ( A ,  B ) such that  F'  ( x )  /  G'  ( x )  =  ( F ( A )  -  F
( B ) )  /  ( G ( A )  -  G
( B ) ). (We express the condition without division, so that we need no nonzero constraints.) (Contributed by Mario Carneiro, 29-Dec-2016.)
Hypotheses
Ref Expression
cmvth.a  |-  ( ph  ->  A  e.  RR )
cmvth.b  |-  ( ph  ->  B  e.  RR )
cmvth.lt  |-  ( ph  ->  A  <  B )
cmvth.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
cmvth.g  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> RR ) )
cmvth.df  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
cmvth.dg  |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )
Assertion
Ref Expression
cmvth  |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `
 x ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )
Distinct variable groups:    x, A    x, B    x, F    x, G    ph, x

Proof of Theorem cmvth
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 cmvth.a . . 3  |-  ( ph  ->  A  e.  RR )
2 cmvth.b . . 3  |-  ( ph  ->  B  e.  RR )
3 cmvth.lt . . 3  |-  ( ph  ->  A  <  B )
4 eqid 2404 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
54subcn 18849 . . . 4  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
64mulcn 18850 . . . . 5  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
7 cmvth.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 18876 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
97, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> RR )
101rexrd 9090 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR* )
112rexrd 9090 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
121, 2, 3ltled 9177 . . . . . . . . 9  |-  ( ph  ->  A  <_  B )
13 ubicc2 10970 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
1410, 11, 12, 13syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
159, 14ffvelrnd 5830 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
16 lbicc2 10969 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1710, 11, 12, 16syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  A  e.  ( A [,] B ) )
189, 17ffvelrnd 5830 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
1915, 18resubcld 9421 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
20 iccssre 10948 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
211, 2, 20syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
22 ax-resscn 9003 . . . . . . 7  |-  RR  C_  CC
2321, 22syl6ss 3320 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  CC )
2422a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
25 cncfmptc 18894 . . . . . 6  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  e.  RR  /\  ( A [,] B ) 
C_  CC  /\  RR  C_  CC )  ->  ( z  e.  ( A [,] B )  |->  ( ( F `  B )  -  ( F `  A ) ) )  e.  ( ( A [,] B ) -cn-> RR ) )
2619, 23, 24, 25syl3anc 1184 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( F `  B )  -  ( F `  A )
) )  e.  ( ( A [,] B
) -cn-> RR ) )
27 cmvth.g . . . . . . . 8  |-  ( ph  ->  G  e.  ( ( A [,] B )
-cn-> RR ) )
28 cncff 18876 . . . . . . . 8  |-  ( G  e.  ( ( A [,] B ) -cn-> RR )  ->  G :
( A [,] B
) --> RR )
2927, 28syl 16 . . . . . . 7  |-  ( ph  ->  G : ( A [,] B ) --> RR )
3029feqmptd 5738 . . . . . 6  |-  ( ph  ->  G  =  ( z  e.  ( A [,] B )  |->  ( G `
 z ) ) )
3130, 27eqeltrrd 2479 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( G `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
32 remulcl 9031 . . . . 5  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  e.  RR  /\  ( G `  z )  e.  RR )  -> 
( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  e.  RR )
334, 6, 26, 31, 22, 32cncfmpt2ss 18898 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
3429, 14ffvelrnd 5830 . . . . . . 7  |-  ( ph  ->  ( G `  B
)  e.  RR )
3529, 17ffvelrnd 5830 . . . . . . 7  |-  ( ph  ->  ( G `  A
)  e.  RR )
3634, 35resubcld 9421 . . . . . 6  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  RR )
37 cncfmptc 18894 . . . . . 6  |-  ( ( ( ( G `  B )  -  ( G `  A )
)  e.  RR  /\  ( A [,] B ) 
C_  CC  /\  RR  C_  CC )  ->  ( z  e.  ( A [,] B )  |->  ( ( G `  B )  -  ( G `  A ) ) )  e.  ( ( A [,] B ) -cn-> RR ) )
3836, 23, 24, 37syl3anc 1184 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( G `  B )  -  ( G `  A )
) )  e.  ( ( A [,] B
) -cn-> RR ) )
399feqmptd 5738 . . . . . 6  |-  ( ph  ->  F  =  ( z  e.  ( A [,] B )  |->  ( F `
 z ) ) )
4039, 7eqeltrrd 2479 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( F `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
41 remulcl 9031 . . . . 5  |-  ( ( ( ( G `  B )  -  ( G `  A )
)  e.  RR  /\  ( F `  z )  e.  RR )  -> 
( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) )  e.  RR )
424, 6, 38, 40, 22, 41cncfmpt2ss 18898 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
43 resubcl 9321 . . . 4  |-  ( ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  e.  RR  /\  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  e.  RR )  -> 
( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) )  e.  RR )
444, 5, 33, 42, 22, 43cncfmpt2ss 18898 . . 3  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) )  e.  ( ( A [,] B
) -cn-> RR ) )
4519recnd 9070 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
4645adantr 452 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
4729ffvelrnda 5829 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  RR )
4847recnd 9070 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  CC )
4946, 48mulcld 9064 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
5036adantr 452 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  RR )
519ffvelrnda 5829 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
5250, 51remulcld 9072 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  RR )
5352recnd 9070 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
5449, 53subcld 9367 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e.  CC )
554tgioo2 18787 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
56 iccntr 18805 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
571, 2, 56syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5824, 21, 54, 55, 4, 57dvmptntr 19810 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) ) ) )  =  ( RR 
_D  ( z  e.  ( A (,) B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) ) ) ) )
59 reex 9037 . . . . . . . . 9  |-  RR  e.  _V
6059prid1 3872 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
6160a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
62 ioossicc 10952 . . . . . . . . 9  |-  ( A (,) B )  C_  ( A [,] B )
6362sseli 3304 . . . . . . . 8  |-  ( z  e.  ( A (,) B )  ->  z  e.  ( A [,] B
) )
6463, 49sylan2 461 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
65 ovex 6065 . . . . . . . 8  |-  ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  e. 
_V
6665a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  e. 
_V )
6763, 48sylan2 461 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( G `  z )  e.  CC )
68 fvex 5701 . . . . . . . . 9  |-  ( ( RR  _D  G ) `
 z )  e. 
_V
6968a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  z )  e.  _V )
7030oveq2d 6056 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( G `  z ) ) ) )
71 dvf 19747 . . . . . . . . . . 11  |-  ( RR 
_D  G ) : dom  ( RR  _D  G ) --> CC
72 cmvth.dg . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )
7372feq2d 5540 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  G ) : dom  ( RR  _D  G
) --> CC  <->  ( RR  _D  G ) : ( A (,) B ) --> CC ) )
7471, 73mpbii 203 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  G
) : ( A (,) B ) --> CC )
7574feqmptd 5738 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G ) `
 z ) ) )
7624, 21, 48, 55, 4, 57dvmptntr 19810 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( G `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( G `
 z ) ) ) )
7770, 75, 763eqtr3rd 2445 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( G `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G
) `  z )
) )
7861, 67, 69, 77, 45dvmptcmul 19803 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) ) ) )
7963, 53sylan2 461 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
80 ovex 6065 . . . . . . . 8  |-  ( ( ( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  z ) )  e. 
_V
8180a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  z ) )  e. 
_V )
8251recnd 9070 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  CC )
8363, 82sylan2 461 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( F `  z )  e.  CC )
84 fvex 5701 . . . . . . . . 9  |-  ( ( RR  _D  F ) `
 z )  e. 
_V
8584a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  _V )
8639oveq2d 6056 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( F `  z ) ) ) )
87 dvf 19747 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
88 cmvth.df . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
8988feq2d 5540 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
9087, 89mpbii 203 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
9190feqmptd 5738 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 z ) ) )
9224, 21, 82, 55, 4, 57dvmptntr 19810 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( F `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( F `
 z ) ) ) )
9386, 91, 923eqtr3rd 2445 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( F `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  z )
) )
9436recnd 9070 . . . . . . . 8  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  CC )
9561, 83, 85, 93, 94dvmptcmul 19803 . . . . . . 7  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( ( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) ) ) )
9661, 64, 66, 78, 79, 81, 95dvmptsub 19806 . . . . . 6  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) ) ) )
9758, 96eqtrd 2436 . . . . 5  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) ) ) )
9897dmeqd 5031 . . . 4  |-  ( ph  ->  dom  ( RR  _D  ( z  e.  ( A [,] B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) ) )  =  dom  ( z  e.  ( A (,) B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `
 z ) ) ) ) )
99 ovex 6065 . . . . 5  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) )  e. 
_V
100 eqid 2404 . . . . 5  |-  ( z  e.  ( A (,) B )  |->  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) ) )  =  ( z  e.  ( A (,) B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `
 z ) ) ) )
10199, 100dmmpti 5533 . . . 4  |-  dom  (
z  e.  ( A (,) B )  |->  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) ) )  =  ( A (,) B )
10298, 101syl6eq 2452 . . 3  |-  ( ph  ->  dom  ( RR  _D  ( z  e.  ( A [,] B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) ) )  =  ( A (,) B
) )
10315recnd 9070 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  CC )
10435recnd 9070 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  e.  CC )
105103, 104mulcld 9064 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  A )
)  e.  CC )
10618recnd 9070 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  CC )
10734recnd 9070 . . . . . . . 8  |-  ( ph  ->  ( G `  B
)  e.  CC )
108106, 107mulcld 9064 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  B )
)  e.  CC )
109106, 104mulcld 9064 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  A )
)  e.  CC )
110105, 108, 109nnncan2d 9402 . . . . . 6  |-  ( ph  ->  ( ( ( ( F `  B )  x.  ( G `  A ) )  -  ( ( F `  A )  x.  ( G `  A )
) )  -  (
( ( F `  A )  x.  ( G `  B )
)  -  ( ( F `  A )  x.  ( G `  A ) ) ) )  =  ( ( ( F `  B
)  x.  ( G `
 A ) )  -  ( ( F `
 A )  x.  ( G `  B
) ) ) )
111103, 107mulcld 9064 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  B )
)  e.  CC )
112111, 108, 105nnncan1d 9401 . . . . . 6  |-  ( ph  ->  ( ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  B )
) )  -  (
( ( F `  B )  x.  ( G `  B )
)  -  ( ( F `  B )  x.  ( G `  A ) ) ) )  =  ( ( ( F `  B
)  x.  ( G `
 A ) )  -  ( ( F `
 A )  x.  ( G `  B
) ) ) )
113110, 112eqtr4d 2439 . . . . 5  |-  ( ph  ->  ( ( ( ( F `  B )  x.  ( G `  A ) )  -  ( ( F `  A )  x.  ( G `  A )
) )  -  (
( ( F `  A )  x.  ( G `  B )
)  -  ( ( F `  A )  x.  ( G `  A ) ) ) )  =  ( ( ( ( F `  B )  x.  ( G `  B )
)  -  ( ( F `  A )  x.  ( G `  B ) ) )  -  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  B )  x.  ( G `  A )
) ) ) )
114103, 106, 104subdird 9446 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 A ) )  =  ( ( ( F `  B )  x.  ( G `  A ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
11594, 106mulcomd 9065 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( F `
 A )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
116106, 107, 104subdid 9445 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  A
)  x.  ( G `
 B ) )  -  ( ( F `
 A )  x.  ( G `  A
) ) ) )
117115, 116eqtrd 2436 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( ( F `  A )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
118114, 117oveq12d 6058 . . . . 5  |-  ( ph  ->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 A ) ) )  =  ( ( ( ( F `  B )  x.  ( G `  A )
)  -  ( ( F `  A )  x.  ( G `  A ) ) )  -  ( ( ( F `  A )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) ) )
119103, 106, 107subdird 9446 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  B )
) ) )
12094, 103mulcomd 9065 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( F `
 B )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
121103, 107, 104subdid 9445 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  B
)  x.  ( G `
 B ) )  -  ( ( F `
 B )  x.  ( G `  A
) ) ) )
122120, 121eqtrd 2436 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  B )  x.  ( G `  A )
) ) )
123119, 122oveq12d 6058 . . . . 5  |-  ( ph  ->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  B
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 B ) ) )  =  ( ( ( ( F `  B )  x.  ( G `  B )
)  -  ( ( F `  A )  x.  ( G `  B ) ) )  -  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  B )  x.  ( G `  A )
) ) ) )
124113, 118, 1233eqtr4d 2446 . . . 4  |-  ( ph  ->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 A ) ) )  =  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 B ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) ) )
125 fveq2 5687 . . . . . . . 8  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
126125oveq2d 6056 . . . . . . 7  |-  ( z  =  A  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) ) )
127 fveq2 5687 . . . . . . . 8  |-  ( z  =  A  ->  ( F `  z )  =  ( F `  A ) )
128127oveq2d 6056 . . . . . . 7  |-  ( z  =  A  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  A
) ) )
129126, 128oveq12d 6058 . . . . . 6  |-  ( z  =  A  ->  (
( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  =  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 A ) ) ) )
130 eqid 2404 . . . . . 6  |-  ( z  e.  ( A [,] B )  |->  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) ) )  =  ( z  e.  ( A [,] B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) ) )
131 ovex 6065 . . . . . 6  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e. 
_V
132129, 130, 131fvmpt3i 5768 . . . . 5  |-  ( A  e.  ( A [,] B )  ->  (
( z  e.  ( A [,] B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) ) `  A
)  =  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 A ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  A
) ) ) )
13317, 132syl 16 . . . 4  |-  ( ph  ->  ( ( z  e.  ( A [,] B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) ) ) `  A
)  =  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 A ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  A
) ) ) )
134 fveq2 5687 . . . . . . . 8  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
135134oveq2d 6056 . . . . . . 7  |-  ( z  =  B  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  B
) ) )
136 fveq2 5687 . . . . . . . 8  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
137136oveq2d 6056 . . . . . . 7  |-  ( z  =  B  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) )
138135, 137oveq12d 6058 . . . . . 6  |-  ( z  =  B  ->  (
( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  =  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  B
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 B ) ) ) )
139138, 130, 131fvmpt3i 5768 . . . . 5  |-  ( B  e.  ( A [,] B )  ->  (
( z  e.  ( A [,] B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) ) `  B
)  =  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 B ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) ) )
14014, 139syl 16 . . . 4  |-  ( ph  ->  ( ( z  e.  ( A [,] B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) ) ) `  B
)  =  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 B ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) ) )
141124, 133, 1403eqtr4d 2446 . . 3  |-  ( ph  ->  ( ( z  e.  ( A [,] B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) ) ) `  A
)  =  ( ( z  e.  ( A [,] B )  |->  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) ) ) `
 B ) )
1421, 2, 3, 44, 102, 141rolle 19827 . 2  |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( RR  _D  ( z  e.  ( A [,] B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) ) ) `  x )  =  0 )
14397fveq1d 5689 . . . . . 6  |-  ( ph  ->  ( ( RR  _D  ( z  e.  ( A [,] B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) ) ) ) ) `  x )  =  ( ( z  e.  ( A (,) B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) ) ) ) `  x
) )
144 fveq2 5687 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  G
) `  z )  =  ( ( RR 
_D  G ) `  x ) )
145144oveq2d 6056 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  x
) ) )
146 fveq2 5687 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  F
) `  z )  =  ( ( RR 
_D  F ) `  x ) )
147146oveq2d 6056 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )
148145, 147oveq12d 6058 . . . . . . 7  |-  ( z  =  x  ->  (
( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) )  =  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  x
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 x ) ) ) )
149148, 100, 99fvmpt3i 5768 . . . . . 6  |-  ( x  e.  ( A (,) B )  ->  (
( z  e.  ( A (,) B ) 
|->  ( ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  z
) )  -  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) ) ) ) `  x
)  =  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 x ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) ) )
150143, 149sylan9eq 2456 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  ( z  e.  ( A [,] B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) ) ) ) `  x )  =  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `
 x ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) ) )
151150eqeq1d 2412 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  (
z  e.  ( A [,] B )  |->  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) ) ) ) `  x )  =  0  <->  ( (
( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 x ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )  =  0 ) )
15245adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
15374ffvelrnda 5829 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  x )  e.  CC )
154152, 153mulcld 9064 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  x ) )  e.  CC )
15594adantr 452 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  CC )
15690ffvelrnda 5829 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
157155, 156mulcld 9064 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  e.  CC )
158154, 157subeq0ad 9377 . . . 4  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `
 x ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )  =  0  <->  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  x
) )  =  ( ( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 x ) ) ) )
159151, 158bitrd 245 . . 3  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( RR  _D  (
z  e.  ( A [,] B )  |->  ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) ) ) ) `  x )  =  0  <->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  x ) )  =  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `
 x ) ) ) )
160159rexbidva 2683 . 2  |-  ( ph  ->  ( E. x  e.  ( A (,) B
) ( ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  -  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) ) ) ) `  x )  =  0  <->  E. x  e.  ( A (,) B ) ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 x ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) ) )
161142, 160mpbid 202 1  |-  ( ph  ->  E. x  e.  ( A (,) B ) ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `
 x ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   _Vcvv 2916    C_ wss 3280   {cpr 3775   class class class wbr 4172    e. cmpt 4226   dom cdm 4837   ran crn 4838   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   RRcr 8945   0cc0 8946    x. cmul 8951   RR*cxr 9075    < clt 9076    <_ cle 9077    - cmin 9247   (,)cioo 10872   [,]cicc 10875   TopOpenctopn 13604   topGenctg 13620  ℂfldccnfld 16658   intcnt 17036   -cn->ccncf 18859    _D cdv 19703
This theorem is referenced by:  mvth  19829  lhop1lem  19850
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-oi 7435  df-card 7782  df-cda 8004  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-ioo 10876  df-ico 10878  df-icc 10879  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-rest 13605  df-topn 13606  df-topgen 13622  df-pt 13623  df-prds 13626  df-xrs 13681  df-0g 13682  df-gsum 13683  df-qtop 13688  df-imas 13689  df-xps 13691  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-submnd 14694  df-mulg 14770  df-cntz 15071  df-cmn 15369  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-fbas 16654  df-fg 16655  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155  df-perf 17156  df-cn 17245  df-cnp 17246  df-haus 17333  df-cmp 17404  df-tx 17547  df-hmeo 17740  df-fil 17831  df-fm 17923  df-flim 17924  df-flf 17925  df-xms 18303  df-ms 18304  df-tms 18305  df-cncf 18861  df-limc 19706  df-dv 19707
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