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Theorem cmvth 22684
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 2402 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
54subcn 21662 . . . 4  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
64mulcn 21663 . . . . 5  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
7 cmvth.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 21689 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
97, 8syl 17 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> RR )
101rexrd 9673 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR* )
112rexrd 9673 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
121, 2, 3ltled 9765 . . . . . . . . 9  |-  ( ph  ->  A  <_  B )
13 ubicc2 11691 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
1410, 11, 12, 13syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
159, 14ffvelrnd 6010 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
16 lbicc2 11690 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1710, 11, 12, 16syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  A  e.  ( A [,] B ) )
189, 17ffvelrnd 6010 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
1915, 18resubcld 10028 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
20 iccssre 11660 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
211, 2, 20syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
22 ax-resscn 9579 . . . . . . 7  |-  RR  C_  CC
2321, 22syl6ss 3454 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  CC )
2422a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
25 cncfmptc 21707 . . . . . 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 1230 . . . . 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 21689 . . . . . . . 8  |-  ( G  e.  ( ( A [,] B ) -cn-> RR )  ->  G :
( A [,] B
) --> RR )
2927, 28syl 17 . . . . . . 7  |-  ( ph  ->  G : ( A [,] B ) --> RR )
3029feqmptd 5902 . . . . . 6  |-  ( ph  ->  G  =  ( z  e.  ( A [,] B )  |->  ( G `
 z ) ) )
3130, 27eqeltrrd 2491 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( G `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
32 remulcl 9607 . . . . 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 21711 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
3429, 14ffvelrnd 6010 . . . . . . 7  |-  ( ph  ->  ( G `  B
)  e.  RR )
3529, 17ffvelrnd 6010 . . . . . . 7  |-  ( ph  ->  ( G `  A
)  e.  RR )
3634, 35resubcld 10028 . . . . . 6  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  RR )
37 cncfmptc 21707 . . . . . 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 1230 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( G `  B )  -  ( G `  A )
) )  e.  ( ( A [,] B
) -cn-> RR ) )
399feqmptd 5902 . . . . . 6  |-  ( ph  ->  F  =  ( z  e.  ( A [,] B )  |->  ( F `
 z ) ) )
4039, 7eqeltrrd 2491 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( F `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
41 remulcl 9607 . . . . 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 21711 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
43 resubcl 9919 . . . 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 21711 . . 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 9652 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
4645adantr 463 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
4729ffvelrnda 6009 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  RR )
4847recnd 9652 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  CC )
4946, 48mulcld 9646 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
5036adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  RR )
519ffvelrnda 6009 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
5250, 51remulcld 9654 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  RR )
5352recnd 9652 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
5449, 53subcld 9967 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e.  CC )
554tgioo2 21600 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
56 iccntr 21618 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
571, 2, 56syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5824, 21, 54, 55, 4, 57dvmptntr 22666 . . . . . 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 reelprrecn 9614 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
6059a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
61 ioossicc 11664 . . . . . . . . 9  |-  ( A (,) B )  C_  ( A [,] B )
6261sseli 3438 . . . . . . . 8  |-  ( z  e.  ( A (,) B )  ->  z  e.  ( A [,] B
) )
6362, 49sylan2 472 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
64 ovex 6306 . . . . . . . 8  |-  ( ( ( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  e. 
_V
6564a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  z ) )  e. 
_V )
6662, 48sylan2 472 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( G `  z )  e.  CC )
67 fvex 5859 . . . . . . . . 9  |-  ( ( RR  _D  G ) `
 z )  e. 
_V
6867a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  z )  e.  _V )
6930oveq2d 6294 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( G `  z ) ) ) )
70 dvf 22603 . . . . . . . . . . 11  |-  ( RR 
_D  G ) : dom  ( RR  _D  G ) --> CC
71 cmvth.dg . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )
7271feq2d 5701 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  G ) : dom  ( RR  _D  G
) --> CC  <->  ( RR  _D  G ) : ( A (,) B ) --> CC ) )
7370, 72mpbii 211 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  G
) : ( A (,) B ) --> CC )
7473feqmptd 5902 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G ) `
 z ) ) )
7524, 21, 48, 55, 4, 57dvmptntr 22666 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( G `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( G `
 z ) ) ) )
7669, 74, 753eqtr3rd 2452 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( G `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G
) `  z )
) )
7760, 66, 68, 76, 45dvmptcmul 22659 . . . . . . 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 ) ) ) )
7862, 53sylan2 472 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
79 ovex 6306 . . . . . . . 8  |-  ( ( ( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  z ) )  e. 
_V
8079a1i 11 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  z ) )  e. 
_V )
8151recnd 9652 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  CC )
8262, 81sylan2 472 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( F `  z )  e.  CC )
83 fvex 5859 . . . . . . . . 9  |-  ( ( RR  _D  F ) `
 z )  e. 
_V
8483a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  _V )
8539oveq2d 6294 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( F `  z ) ) ) )
86 dvf 22603 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
87 cmvth.df . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
8887feq2d 5701 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
8986, 88mpbii 211 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
9089feqmptd 5902 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 z ) ) )
9124, 21, 81, 55, 4, 57dvmptntr 22666 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( F `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( F `
 z ) ) ) )
9285, 90, 913eqtr3rd 2452 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( F `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  z )
) )
9336recnd 9652 . . . . . . . 8  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  CC )
9460, 82, 84, 92, 93dvmptcmul 22659 . . . . . . 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 ) ) ) )
9560, 63, 65, 77, 78, 80, 94dvmptsub 22662 . . . . . 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
) ) ) ) )
9658, 95eqtrd 2443 . . . . 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
) ) ) ) )
9796dmeqd 5026 . . . 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 ) ) ) ) )
98 ovex 6306 . . . . 5  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) )  e. 
_V
99 eqid 2402 . . . . 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 ) ) ) )
10098, 99dmmpti 5693 . . . 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 )
10197, 100syl6eq 2459 . . 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
) )
10215recnd 9652 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  CC )
10335recnd 9652 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  e.  CC )
104102, 103mulcld 9646 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  A )
)  e.  CC )
10518recnd 9652 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  CC )
10634recnd 9652 . . . . . . . 8  |-  ( ph  ->  ( G `  B
)  e.  CC )
107105, 106mulcld 9646 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  B )
)  e.  CC )
108105, 103mulcld 9646 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  A )
)  e.  CC )
109104, 107, 108nnncan2d 10002 . . . . . 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
) ) ) )
110102, 106mulcld 9646 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  B )
)  e.  CC )
111110, 107, 104nnncan1d 10001 . . . . . 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
) ) ) )
112109, 111eqtr4d 2446 . . . . 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 )
) ) ) )
113102, 105, 103subdird 10054 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 A ) )  =  ( ( ( F `  B )  x.  ( G `  A ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
11493, 105mulcomd 9647 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( F `
 A )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
115105, 106, 103subdid 10053 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  A
)  x.  ( G `
 B ) )  -  ( ( F `
 A )  x.  ( G `  A
) ) ) )
116114, 115eqtrd 2443 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( ( F `  A )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
117113, 116oveq12d 6296 . . . . 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 )
) ) ) )
118102, 105, 106subdird 10054 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  B )
) ) )
11993, 102mulcomd 9647 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( F `
 B )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
120102, 106, 103subdid 10053 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  B
)  x.  ( G `
 B ) )  -  ( ( F `
 B )  x.  ( G `  A
) ) ) )
121119, 120eqtrd 2443 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  B )  x.  ( G `  A )
) ) )
122118, 121oveq12d 6296 . . . . 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 )
) ) ) )
123112, 117, 1223eqtr4d 2453 . . . 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
) ) ) )
124 fveq2 5849 . . . . . . . 8  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
125124oveq2d 6294 . . . . . . 7  |-  ( z  =  A  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) ) )
126 fveq2 5849 . . . . . . . 8  |-  ( z  =  A  ->  ( F `  z )  =  ( F `  A ) )
127126oveq2d 6294 . . . . . . 7  |-  ( z  =  A  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  A
) ) )
128125, 127oveq12d 6296 . . . . . 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 ) ) ) )
129 eqid 2402 . . . . . 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 ) ) ) )
130 ovex 6306 . . . . . 6  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e. 
_V
131128, 129, 130fvmpt3i 5937 . . . . 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
) ) ) )
13217, 131syl 17 . . . 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
) ) ) )
133 fveq2 5849 . . . . . . . 8  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
134133oveq2d 6294 . . . . . . 7  |-  ( z  =  B  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  B
) ) )
135 fveq2 5849 . . . . . . . 8  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
136135oveq2d 6294 . . . . . . 7  |-  ( z  =  B  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) )
137134, 136oveq12d 6296 . . . . . 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 ) ) ) )
138137, 129, 130fvmpt3i 5937 . . . . 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
) ) ) )
13914, 138syl 17 . . . 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
) ) ) )
140123, 132, 1393eqtr4d 2453 . . 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 ) )
1411, 2, 3, 44, 101, 140rolle 22683 . 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 )
14296fveq1d 5851 . . . . . 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
) )
143 fveq2 5849 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  G
) `  z )  =  ( ( RR 
_D  G ) `  x ) )
144143oveq2d 6294 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  x
) ) )
145 fveq2 5849 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  F
) `  z )  =  ( ( RR 
_D  F ) `  x ) )
146145oveq2d 6294 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )
147144, 146oveq12d 6296 . . . . . . 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 ) ) ) )
148147, 99, 98fvmpt3i 5937 . . . . . 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
) ) ) )
149142, 148sylan9eq 2463 . . . . 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
) ) ) )
150149eqeq1d 2404 . . . 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 ) )
15145adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
15273ffvelrnda 6009 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  x )  e.  CC )
153151, 152mulcld 9646 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  x ) )  e.  CC )
15493adantr 463 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  CC )
15589ffvelrnda 6009 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
156154, 155mulcld 9646 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  e.  CC )
157153, 156subeq0ad 9977 . . . 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 ) ) ) )
158150, 157bitrd 253 . . 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 ) ) ) )
159158rexbidva 2915 . 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
) ) ) )
160141, 159mpbid 210 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2755   _Vcvv 3059    C_ wss 3414   {cpr 3974   class class class wbr 4395    |-> cmpt 4453   dom cdm 4823   ran crn 4824   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   RRcr 9521   0cc0 9522    x. cmul 9527   RR*cxr 9657    < clt 9658    <_ cle 9659    - cmin 9841   (,)cioo 11582   [,]cicc 11585   TopOpenctopn 15036   topGenctg 15052  ℂfldccnfld 18740   intcnt 19810   -cn->ccncf 21672    _D cdv 22559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-ico 11588  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-fbas 18736  df-fg 18737  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-ntr 19813  df-cls 19814  df-nei 19892  df-lp 19930  df-perf 19931  df-cn 20021  df-cnp 20022  df-haus 20109  df-cmp 20180  df-tx 20355  df-hmeo 20548  df-fil 20639  df-fm 20731  df-flim 20732  df-flf 20733  df-xms 21115  df-ms 21116  df-tms 21117  df-cncf 21674  df-limc 22562  df-dv 22563
This theorem is referenced by:  mvth  22685  lhop1lem  22706
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