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Theorem cmvth 19354
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 2296 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
54subcn 18386 . . . 4  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
64mulcn 18387 . . . . 5  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
7 cmvth.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 18413 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
97, 8syl 15 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> RR )
101rexrd 8897 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR* )
112rexrd 8897 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
121, 2, 3ltled 8983 . . . . . . . . 9  |-  ( ph  ->  A  <_  B )
13 ubicc2 10769 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
1410, 11, 12, 13syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
159, 14ffvelrnd 5682 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
16 lbicc2 10768 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1710, 11, 12, 16syl3anc 1182 . . . . . . . 8  |-  ( ph  ->  A  e.  ( A [,] B ) )
189, 17ffvelrnd 5682 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
1915, 18resubcld 9227 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
20 iccssre 10747 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
211, 2, 20syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
22 ax-resscn 8810 . . . . . . 7  |-  RR  C_  CC
2321, 22syl6ss 3204 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  CC )
2422a1i 10 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
25 cncfmptc 18431 . . . . . 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 1182 . . . . 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 18413 . . . . . . . 8  |-  ( G  e.  ( ( A [,] B ) -cn-> RR )  ->  G :
( A [,] B
) --> RR )
2927, 28syl 15 . . . . . . 7  |-  ( ph  ->  G : ( A [,] B ) --> RR )
3029feqmptd 5591 . . . . . 6  |-  ( ph  ->  G  =  ( z  e.  ( A [,] B )  |->  ( G `
 z ) ) )
3130, 27eqeltrrd 2371 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( G `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
32 remulcl 8838 . . . . 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 18435 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
3429, 14ffvelrnd 5682 . . . . . . 7  |-  ( ph  ->  ( G `  B
)  e.  RR )
3529, 17ffvelrnd 5682 . . . . . . 7  |-  ( ph  ->  ( G `  A
)  e.  RR )
3634, 35resubcld 9227 . . . . . 6  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  RR )
37 cncfmptc 18431 . . . . . 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 1182 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( G `  B )  -  ( G `  A )
) )  e.  ( ( A [,] B
) -cn-> RR ) )
399feqmptd 5591 . . . . . 6  |-  ( ph  ->  F  =  ( z  e.  ( A [,] B )  |->  ( F `
 z ) ) )
4039, 7eqeltrrd 2371 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( F `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
41 remulcl 8838 . . . . 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 18435 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
43 resubcl 9127 . . . 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 18435 . . 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 8877 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
4645adantr 451 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
4729ffvelrnda 5681 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  RR )
4847recnd 8877 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  CC )
4946, 48mulcld 8871 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
5036adantr 451 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  RR )
519ffvelrnda 5681 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
5250, 51remulcld 8879 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  RR )
5352recnd 8877 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
5449, 53subcld 9173 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e.  CC )
554tgioo2 18325 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
56 iccntr 18342 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
571, 2, 56syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5824, 21, 54, 55, 4, 57dvmptntr 19336 . . . . . 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 8844 . . . . . . . . 9  |-  RR  e.  _V
6059prid1 3747 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
6160a1i 10 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
62 ioossicc 10751 . . . . . . . . 9  |-  ( A (,) B )  C_  ( A [,] B )
6362sseli 3189 . . . . . . . 8  |-  ( z  e.  ( A (,) B )  ->  z  e.  ( A [,] B
) )
6463, 49sylan2 460 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
65 ovex 5899 . . . . . . . 8  |-  ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  e. 
_V
6665a1i 10 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  e. 
_V )
6763, 48sylan2 460 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( G `  z )  e.  CC )
68 fvex 5555 . . . . . . . . 9  |-  ( ( RR  _D  G ) `
 z )  e. 
_V
6968a1i 10 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  z )  e.  _V )
7030oveq2d 5890 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( G `  z ) ) ) )
71 dvf 19273 . . . . . . . . . . 11  |-  ( RR 
_D  G ) : dom  ( RR  _D  G ) --> CC
72 cmvth.dg . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )
7372feq2d 5396 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  G ) : dom  ( RR  _D  G
) --> CC  <->  ( RR  _D  G ) : ( A (,) B ) --> CC ) )
7471, 73mpbii 202 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  G
) : ( A (,) B ) --> CC )
7574feqmptd 5591 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G ) `
 z ) ) )
7624, 21, 48, 55, 4, 57dvmptntr 19336 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( G `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( G `
 z ) ) ) )
7770, 75, 763eqtr3rd 2337 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( G `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G
) `  z )
) )
7861, 67, 69, 77, 45dvmptcmul 19329 . . . . . . 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 460 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
80 ovex 5899 . . . . . . . 8  |-  ( ( ( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  z ) )  e. 
_V
8180a1i 10 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  z ) )  e. 
_V )
8251recnd 8877 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  CC )
8363, 82sylan2 460 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( F `  z )  e.  CC )
84 fvex 5555 . . . . . . . . 9  |-  ( ( RR  _D  F ) `
 z )  e. 
_V
8584a1i 10 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  _V )
8639oveq2d 5890 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( F `  z ) ) ) )
87 dvf 19273 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
88 cmvth.df . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
8988feq2d 5396 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
9087, 89mpbii 202 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
9190feqmptd 5591 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 z ) ) )
9224, 21, 82, 55, 4, 57dvmptntr 19336 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( F `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( F `
 z ) ) ) )
9386, 91, 923eqtr3rd 2337 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( F `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  z )
) )
9436recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  CC )
9561, 83, 85, 93, 94dvmptcmul 19329 . . . . . . 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 19332 . . . . . 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 2328 . . . . 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 4897 . . . 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 5899 . . . . 5  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) )  e. 
_V
100 eqid 2296 . . . . 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 5389 . . . 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 2344 . . 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 8877 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  CC )
10435recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  e.  CC )
105103, 104mulcld 8871 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  A )
)  e.  CC )
10618recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  CC )
10734recnd 8877 . . . . . . . 8  |-  ( ph  ->  ( G `  B
)  e.  CC )
108106, 107mulcld 8871 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  B )
)  e.  CC )
109106, 104mulcld 8871 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  A )
)  e.  CC )
110105, 108, 109nnncan2d 9208 . . . . . 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 8871 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  B )
)  e.  CC )
112111, 108, 105nnncan1d 9207 . . . . . 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 2331 . . . . 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 9252 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 A ) )  =  ( ( ( F `  B )  x.  ( G `  A ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
11594, 106mulcomd 8872 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( F `
 A )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
116106, 107, 104subdid 9251 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  A
)  x.  ( G `
 B ) )  -  ( ( F `
 A )  x.  ( G `  A
) ) ) )
117115, 116eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( ( F `  A )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
118114, 117oveq12d 5892 . . . . 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 9252 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  B )
) ) )
12094, 103mulcomd 8872 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( F `
 B )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
121103, 107, 104subdid 9251 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  B
)  x.  ( G `
 B ) )  -  ( ( F `
 B )  x.  ( G `  A
) ) ) )
122120, 121eqtrd 2328 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  B )  x.  ( G `  A )
) ) )
123119, 122oveq12d 5892 . . . . 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 2338 . . . 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 5541 . . . . . . . 8  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
126125oveq2d 5890 . . . . . . 7  |-  ( z  =  A  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) ) )
127 fveq2 5541 . . . . . . . 8  |-  ( z  =  A  ->  ( F `  z )  =  ( F `  A ) )
128127oveq2d 5890 . . . . . . 7  |-  ( z  =  A  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  A
) ) )
129126, 128oveq12d 5892 . . . . . 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 2296 . . . . . 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 5899 . . . . . 6  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e. 
_V
132129, 130, 131fvmpt3i 5621 . . . . 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 15 . . . 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 5541 . . . . . . . 8  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
135134oveq2d 5890 . . . . . . 7  |-  ( z  =  B  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  B
) ) )
136 fveq2 5541 . . . . . . . 8  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
137136oveq2d 5890 . . . . . . 7  |-  ( z  =  B  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) )
138135, 137oveq12d 5892 . . . . . 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 5621 . . . . 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 15 . . . 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 2338 . . 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 19353 . 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 5543 . . . . . 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 5541 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  G
) `  z )  =  ( ( RR 
_D  G ) `  x ) )
145144oveq2d 5890 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  x
) ) )
146 fveq2 5541 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  F
) `  z )  =  ( ( RR 
_D  F ) `  x ) )
147146oveq2d 5890 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )
148145, 147oveq12d 5892 . . . . . . 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 5621 . . . . . 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 2348 . . . . 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 2304 . . . 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 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
15374ffvelrnda 5681 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  x )  e.  CC )
154152, 153mulcld 8871 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  x ) )  e.  CC )
15594adantr 451 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  CC )
15690ffvelrnda 5681 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
157155, 156mulcld 8871 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  e.  CC )
158 subeq0 9089 . . . . 5  |-  ( ( ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `
 x ) )  e.  CC  /\  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 x ) )  e.  CC )  -> 
( ( ( ( ( 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
) ) ) )
159154, 157, 158syl2anc 642 . . . 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 ) ) ) )
160151, 159bitrd 244 . . 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 ) ) ) )
161160rexbidva 2573 . 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
) ) ) )
162142, 161mpbid 201 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    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   _Vcvv 2801    C_ wss 3165   {cpr 3654   class class class wbr 4039    e. cmpt 4093   dom cdm 4705   ran crn 4706   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   RRcr 8752   0cc0 8753    x. cmul 8758   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   (,)cioo 10672   [,]cicc 10675   TopOpenctopn 13342   topGenctg 13358  ℂfldccnfld 16393   intcnt 16770   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  mvth  19355  lhop1lem  19376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-ixp 6834  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-fi 7181  df-sup 7210  df-oi 7241  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-uz 10247  df-q 10333  df-rp 10371  df-xneg 10468  df-xadd 10469  df-xmul 10470  df-ioo 10676  df-ico 10678  df-icc 10679  df-fz 10799  df-fzo 10887  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-struct 13166  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-starv 13239  df-sca 13240  df-vsca 13241  df-tset 13243  df-ple 13244  df-ds 13246  df-hom 13248  df-cco 13249  df-rest 13343  df-topn 13344  df-topgen 13360  df-pt 13361  df-prds 13364  df-xrs 13419  df-0g 13420  df-gsum 13421  df-qtop 13426  df-imas 13427  df-xps 13429  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-mulg 14508  df-cntz 14809  df-cmn 15107  df-xmet 16389  df-met 16390  df-bl 16391  df-mopn 16392  df-cnfld 16394  df-top 16652  df-bases 16654  df-topon 16655  df-topsp 16656  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884  df-perf 16885  df-cn 16973  df-cnp 16974  df-haus 17059  df-cmp 17130  df-tx 17273  df-hmeo 17462  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651  df-xms 17901  df-ms 17902  df-tms 17903  df-cncf 18398  df-limc 19232  df-dv 19233
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