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Theorem cmvth 23022
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 2471 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
54subcn 21976 . . . 4  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
64mulcn 21977 . . . . 5  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
7 cmvth.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 22003 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
97, 8syl 17 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> RR )
101rexrd 9708 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR* )
112rexrd 9708 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
121, 2, 3ltled 9800 . . . . . . . . 9  |-  ( ph  ->  A  <_  B )
13 ubicc2 11775 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
1410, 11, 12, 13syl3anc 1292 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
159, 14ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
16 lbicc2 11774 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1710, 11, 12, 16syl3anc 1292 . . . . . . . 8  |-  ( ph  ->  A  e.  ( A [,] B ) )
189, 17ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
1915, 18resubcld 10068 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
20 iccssre 11741 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
211, 2, 20syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
22 ax-resscn 9614 . . . . . . 7  |-  RR  C_  CC
2321, 22syl6ss 3430 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  CC )
2422a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
25 cncfmptc 22021 . . . . . 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 1292 . . . . 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 22003 . . . . . . . 8  |-  ( G  e.  ( ( A [,] B ) -cn-> RR )  ->  G :
( A [,] B
) --> RR )
2927, 28syl 17 . . . . . . 7  |-  ( ph  ->  G : ( A [,] B ) --> RR )
3029feqmptd 5932 . . . . . 6  |-  ( ph  ->  G  =  ( z  e.  ( A [,] B )  |->  ( G `
 z ) ) )
3130, 27eqeltrrd 2550 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( G `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
32 remulcl 9642 . . . . 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 22025 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
3429, 14ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( G `  B
)  e.  RR )
3529, 17ffvelrnd 6038 . . . . . . 7  |-  ( ph  ->  ( G `  A
)  e.  RR )
3634, 35resubcld 10068 . . . . . 6  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  RR )
37 cncfmptc 22021 . . . . . 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 1292 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( G `  B )  -  ( G `  A )
) )  e.  ( ( A [,] B
) -cn-> RR ) )
399feqmptd 5932 . . . . . 6  |-  ( ph  ->  F  =  ( z  e.  ( A [,] B )  |->  ( F `
 z ) ) )
4039, 7eqeltrrd 2550 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( F `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
41 remulcl 9642 . . . . 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 22025 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
43 resubcl 9958 . . . 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 22025 . . 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 9687 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
4645adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
4729ffvelrnda 6037 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  RR )
4847recnd 9687 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  CC )
4946, 48mulcld 9681 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
5036adantr 472 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  RR )
519ffvelrnda 6037 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
5250, 51remulcld 9689 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  RR )
5352recnd 9687 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
5449, 53subcld 10005 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e.  CC )
554tgioo2 21899 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
56 iccntr 21917 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
571, 2, 56syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5824, 21, 54, 55, 4, 57dvmptntr 23004 . . . . . 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 9649 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
6059a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
61 ioossicc 11745 . . . . . . . . 9  |-  ( A (,) B )  C_  ( A [,] B )
6261sseli 3414 . . . . . . . 8  |-  ( z  e.  ( A (,) B )  ->  z  e.  ( A [,] B
) )
6362, 49sylan2 482 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
64 ovex 6336 . . . . . . . 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 482 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( G `  z )  e.  CC )
67 fvex 5889 . . . . . . . . 9  |-  ( ( RR  _D  G ) `
 z )  e. 
_V
6867a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  z )  e.  _V )
6930oveq2d 6324 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( G `  z ) ) ) )
70 dvf 22941 . . . . . . . . . . 11  |-  ( RR 
_D  G ) : dom  ( RR  _D  G ) --> CC
71 cmvth.dg . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )
7271feq2d 5725 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  G ) : dom  ( RR  _D  G
) --> CC  <->  ( RR  _D  G ) : ( A (,) B ) --> CC ) )
7370, 72mpbii 216 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  G
) : ( A (,) B ) --> CC )
7473feqmptd 5932 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G ) `
 z ) ) )
7524, 21, 48, 55, 4, 57dvmptntr 23004 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( G `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( G `
 z ) ) ) )
7669, 74, 753eqtr3rd 2514 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( G `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G
) `  z )
) )
7760, 66, 68, 76, 45dvmptcmul 22997 . . . . . . 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 482 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
79 ovex 6336 . . . . . . . 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 9687 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  CC )
8262, 81sylan2 482 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( F `  z )  e.  CC )
83 fvex 5889 . . . . . . . . 9  |-  ( ( RR  _D  F ) `
 z )  e. 
_V
8483a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  _V )
8539oveq2d 6324 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( F `  z ) ) ) )
86 dvf 22941 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
87 cmvth.df . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
8887feq2d 5725 . . . . . . . . . . 11  |-  ( ph  ->  ( ( RR  _D  F ) : dom  ( RR  _D  F
) --> CC  <->  ( RR  _D  F ) : ( A (,) B ) --> CC ) )
8986, 88mpbii 216 . . . . . . . . . 10  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> CC )
9089feqmptd 5932 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 z ) ) )
9124, 21, 81, 55, 4, 57dvmptntr 23004 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( F `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( F `
 z ) ) ) )
9285, 90, 913eqtr3rd 2514 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( F `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  z )
) )
9336recnd 9687 . . . . . . . 8  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  CC )
9460, 82, 84, 92, 93dvmptcmul 22997 . . . . . . 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 23000 . . . . . 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 2505 . . . . 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 5042 . . . 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 6336 . . . . 5  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) )  e. 
_V
99 eqid 2471 . . . . 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 5717 . . . 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 2521 . . 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 9687 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  CC )
10335recnd 9687 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  e.  CC )
104102, 103mulcld 9681 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  A )
)  e.  CC )
10518recnd 9687 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  CC )
10634recnd 9687 . . . . . . . 8  |-  ( ph  ->  ( G `  B
)  e.  CC )
107105, 106mulcld 9681 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  B )
)  e.  CC )
108105, 103mulcld 9681 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  A )
)  e.  CC )
109104, 107, 108nnncan2d 10040 . . . . . 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 9681 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  B )
)  e.  CC )
111110, 107, 104nnncan1d 10039 . . . . . 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 2508 . . . . 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 10096 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 A ) )  =  ( ( ( F `  B )  x.  ( G `  A ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
11493, 105mulcomd 9682 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( F `
 A )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
115105, 106, 103subdid 10095 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  A
)  x.  ( G `
 B ) )  -  ( ( F `
 A )  x.  ( G `  A
) ) ) )
116114, 115eqtrd 2505 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( ( F `  A )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
117113, 116oveq12d 6326 . . . . 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 10096 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  B )
) ) )
11993, 102mulcomd 9682 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( F `
 B )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
120102, 106, 103subdid 10095 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  B
)  x.  ( G `
 B ) )  -  ( ( F `
 B )  x.  ( G `  A
) ) ) )
121119, 120eqtrd 2505 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  B )  x.  ( G `  A )
) ) )
122118, 121oveq12d 6326 . . . . 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 2515 . . . 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 5879 . . . . . . . 8  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
125124oveq2d 6324 . . . . . . 7  |-  ( z  =  A  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) ) )
126 fveq2 5879 . . . . . . . 8  |-  ( z  =  A  ->  ( F `  z )  =  ( F `  A ) )
127126oveq2d 6324 . . . . . . 7  |-  ( z  =  A  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  A
) ) )
128125, 127oveq12d 6326 . . . . . 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 2471 . . . . . 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 6336 . . . . . 6  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e. 
_V
131128, 129, 130fvmpt3i 5968 . . . . 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 5879 . . . . . . . 8  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
134133oveq2d 6324 . . . . . . 7  |-  ( z  =  B  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  B
) ) )
135 fveq2 5879 . . . . . . . 8  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
136135oveq2d 6324 . . . . . . 7  |-  ( z  =  B  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) )
137134, 136oveq12d 6326 . . . . . 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 5968 . . . . 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 2515 . . 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 23021 . 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 5881 . . . . . 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 5879 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  G
) `  z )  =  ( ( RR 
_D  G ) `  x ) )
144143oveq2d 6324 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  x
) ) )
145 fveq2 5879 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  F
) `  z )  =  ( ( RR 
_D  F ) `  x ) )
146145oveq2d 6324 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )
147144, 146oveq12d 6326 . . . . . . 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 5968 . . . . . 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 2525 . . . . 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 2473 . . . 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 472 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
15273ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  x )  e.  CC )
153151, 152mulcld 9681 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  x ) )  e.  CC )
15493adantr 472 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  CC )
15589ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
156154, 155mulcld 9681 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  e.  CC )
157153, 156subeq0ad 10015 . . . 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 261 . . 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 2889 . 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 215 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 376    = wceq 1452    e. wcel 1904   E.wrex 2757   _Vcvv 3031    C_ wss 3390   {cpr 3961   class class class wbr 4395    |-> cmpt 4454   dom cdm 4839   ran crn 4840   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557    x. cmul 9562   RR*cxr 9692    < clt 9693    <_ cle 9694    - cmin 9880   (,)cioo 11660   [,]cicc 11663   TopOpenctopn 15398   topGenctg 15414  ℂfldccnfld 19047   intcnt 20109   -cn->ccncf 21986    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ioo 11664  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-haus 20408  df-cmp 20479  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by:  mvth  23023  lhop1lem  23044
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