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Theorem cmvth 22120
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 2460 . . . 4  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
54subcn 21098 . . . 4  |-  -  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
64mulcn 21099 . . . . 5  |-  x.  e.  ( ( ( TopOpen ` fld )  tX  ( TopOpen ` fld ) )  Cn  ( TopOpen
` fld
) )
7 cmvth.f . . . . . . . . 9  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
8 cncff 21125 . . . . . . . . 9  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
97, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : ( A [,] B ) --> RR )
101rexrd 9632 . . . . . . . . 9  |-  ( ph  ->  A  e.  RR* )
112rexrd 9632 . . . . . . . . 9  |-  ( ph  ->  B  e.  RR* )
121, 2, 3ltled 9721 . . . . . . . . 9  |-  ( ph  ->  A  <_  B )
13 ubicc2 11626 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  B  e.  ( A [,] B
) )
1410, 11, 12, 13syl3anc 1223 . . . . . . . 8  |-  ( ph  ->  B  e.  ( A [,] B ) )
159, 14ffvelrnd 6013 . . . . . . 7  |-  ( ph  ->  ( F `  B
)  e.  RR )
16 lbicc2 11625 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
1710, 11, 12, 16syl3anc 1223 . . . . . . . 8  |-  ( ph  ->  A  e.  ( A [,] B ) )
189, 17ffvelrnd 6013 . . . . . . 7  |-  ( ph  ->  ( F `  A
)  e.  RR )
1915, 18resubcld 9976 . . . . . 6  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  RR )
20 iccssre 11595 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
211, 2, 20syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( A [,] B
)  C_  RR )
22 ax-resscn 9538 . . . . . . 7  |-  RR  C_  CC
2321, 22syl6ss 3509 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  CC )
2422a1i 11 . . . . . 6  |-  ( ph  ->  RR  C_  CC )
25 cncfmptc 21143 . . . . . 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 1223 . . . . 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 21125 . . . . . . . 8  |-  ( G  e.  ( ( A [,] B ) -cn-> RR )  ->  G :
( A [,] B
) --> RR )
2927, 28syl 16 . . . . . . 7  |-  ( ph  ->  G : ( A [,] B ) --> RR )
3029feqmptd 5911 . . . . . 6  |-  ( ph  ->  G  =  ( z  e.  ( A [,] B )  |->  ( G `
 z ) ) )
3130, 27eqeltrrd 2549 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( G `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
32 remulcl 9566 . . . . 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 21147 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
3429, 14ffvelrnd 6013 . . . . . . 7  |-  ( ph  ->  ( G `  B
)  e.  RR )
3529, 17ffvelrnd 6013 . . . . . . 7  |-  ( ph  ->  ( G `  A
)  e.  RR )
3634, 35resubcld 9976 . . . . . 6  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  RR )
37 cncfmptc 21143 . . . . . 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 1223 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( G `  B )  -  ( G `  A )
) )  e.  ( ( A [,] B
) -cn-> RR ) )
399feqmptd 5911 . . . . . 6  |-  ( ph  ->  F  =  ( z  e.  ( A [,] B )  |->  ( F `
 z ) ) )
4039, 7eqeltrrd 2549 . . . . 5  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( F `  z
) )  e.  ( ( A [,] B
) -cn-> RR ) )
41 remulcl 9566 . . . . 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 21147 . . . 4  |-  ( ph  ->  ( z  e.  ( A [,] B ) 
|->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 z ) ) )  e.  ( ( A [,] B )
-cn-> RR ) )
43 resubcl 9872 . . . 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 21147 . . 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 9611 . . . . . . . . . 10  |-  ( ph  ->  ( ( F `  B )  -  ( F `  A )
)  e.  CC )
4645adantr 465 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
4729ffvelrnda 6012 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  RR )
4847recnd 9611 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( G `  z )  e.  CC )
4946, 48mulcld 9605 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
5036adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  RR )
519ffvelrnda 6012 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  RR )
5250, 51remulcld 9613 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  RR )
5352recnd 9611 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
5449, 53subcld 9919 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e.  CC )
554tgioo2 21036 . . . . . . 7  |-  ( topGen ` 
ran  (,) )  =  ( ( TopOpen ` fld )t  RR )
56 iccntr 21054 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
571, 2, 56syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( ( int `  ( topGen `
 ran  (,) )
) `  ( A [,] B ) )  =  ( A (,) B
) )
5824, 21, 54, 55, 4, 57dvmptntr 22102 . . . . . 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 9573 . . . . . . . 8  |-  RR  e.  { RR ,  CC }
6059a1i 11 . . . . . . 7  |-  ( ph  ->  RR  e.  { RR ,  CC } )
61 ioossicc 11599 . . . . . . . . 9  |-  ( A (,) B )  C_  ( A [,] B )
6261sseli 3493 . . . . . . . 8  |-  ( z  e.  ( A (,) B )  ->  z  e.  ( A [,] B
) )
6362, 49sylan2 474 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( G `  z ) )  e.  CC )
64 ovex 6300 . . . . . . . 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 474 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( G `  z )  e.  CC )
67 fvex 5867 . . . . . . . . 9  |-  ( ( RR  _D  G ) `
 z )  e. 
_V
6867a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  z )  e.  _V )
6930oveq2d 6291 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( G `  z ) ) ) )
70 dvf 22039 . . . . . . . . . . 11  |-  ( RR 
_D  G ) : dom  ( RR  _D  G ) --> CC
71 cmvth.dg . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  G )  =  ( A (,) B ) )
7271feq2d 5709 . . . . . . . . . . 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 5911 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  G
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G ) `
 z ) ) )
7524, 21, 48, 55, 4, 57dvmptntr 22102 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( G `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( G `
 z ) ) ) )
7669, 74, 753eqtr3rd 2510 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( G `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  G
) `  z )
) )
7760, 66, 68, 76, 45dvmptcmul 22095 . . . . . . 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 474 . . . . . . 7  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( F `  z ) )  e.  CC )
79 ovex 6300 . . . . . . . 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 9611 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  ( A [,] B ) )  ->  ( F `  z )  e.  CC )
8262, 81sylan2 474 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( F `  z )  e.  CC )
83 fvex 5867 . . . . . . . . 9  |-  ( ( RR  _D  F ) `
 z )  e. 
_V
8483a1i 11 . . . . . . . 8  |-  ( (
ph  /\  z  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  z )  e.  _V )
8539oveq2d 6291 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( RR 
_D  ( z  e.  ( A [,] B
)  |->  ( F `  z ) ) ) )
86 dvf 22039 . . . . . . . . . . 11  |-  ( RR 
_D  F ) : dom  ( RR  _D  F ) --> CC
87 cmvth.df . . . . . . . . . . . 12  |-  ( ph  ->  dom  ( RR  _D  F )  =  ( A (,) B ) )
8887feq2d 5709 . . . . . . . . . . 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 5911 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  F
)  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F ) `
 z ) ) )
9124, 21, 81, 55, 4, 57dvmptntr 22102 . . . . . . . . 9  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A [,] B )  |->  ( F `  z ) ) )  =  ( RR  _D  ( z  e.  ( A (,) B )  |->  ( F `
 z ) ) ) )
9285, 90, 913eqtr3rd 2510 . . . . . . . 8  |-  ( ph  ->  ( RR  _D  (
z  e.  ( A (,) B )  |->  ( F `  z ) ) )  =  ( z  e.  ( A (,) B )  |->  ( ( RR  _D  F
) `  z )
) )
9336recnd 9611 . . . . . . . 8  |-  ( ph  ->  ( ( G `  B )  -  ( G `  A )
)  e.  CC )
9460, 82, 84, 92, 93dvmptcmul 22095 . . . . . . 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 22098 . . . . . 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 2501 . . . . 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 5196 . . . 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 6300 . . . . 5  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  z
) ) )  e. 
_V
99 eqid 2460 . . . . 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 5701 . . . 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 2517 . . 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 9611 . . . . . . . 8  |-  ( ph  ->  ( F `  B
)  e.  CC )
10335recnd 9611 . . . . . . . 8  |-  ( ph  ->  ( G `  A
)  e.  CC )
104102, 103mulcld 9605 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  A )
)  e.  CC )
10518recnd 9611 . . . . . . . 8  |-  ( ph  ->  ( F `  A
)  e.  CC )
10634recnd 9611 . . . . . . . 8  |-  ( ph  ->  ( G `  B
)  e.  CC )
107105, 106mulcld 9605 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  B )
)  e.  CC )
108105, 103mulcld 9605 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  ( G `  A )
)  e.  CC )
109104, 107, 108nnncan2d 9954 . . . . . 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 9605 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  ( G `  B )
)  e.  CC )
111110, 107, 104nnncan1d 9953 . . . . . 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 2504 . . . . 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 10002 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 A ) )  =  ( ( ( F `  B )  x.  ( G `  A ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
11493, 105mulcomd 9606 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( F `
 A )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
115105, 106, 103subdid 10001 . . . . . . 7  |-  ( ph  ->  ( ( F `  A )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  A
)  x.  ( G `
 B ) )  -  ( ( F `
 A )  x.  ( G `  A
) ) ) )
116114, 115eqtrd 2501 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 A ) )  =  ( ( ( F `  A )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  A )
) ) )
117113, 116oveq12d 6293 . . . . 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 10002 . . . . . 6  |-  ( ph  ->  ( ( ( F `
 B )  -  ( F `  A ) )  x.  ( G `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  A )  x.  ( G `  B )
) ) )
11993, 102mulcomd 9606 . . . . . . 7  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( F `
 B )  x.  ( ( G `  B )  -  ( G `  A )
) ) )
120102, 106, 103subdid 10001 . . . . . . 7  |-  ( ph  ->  ( ( F `  B )  x.  (
( G `  B
)  -  ( G `
 A ) ) )  =  ( ( ( F `  B
)  x.  ( G `
 B ) )  -  ( ( F `
 B )  x.  ( G `  A
) ) ) )
121119, 120eqtrd 2501 . . . . . 6  |-  ( ph  ->  ( ( ( G `
 B )  -  ( G `  A ) )  x.  ( F `
 B ) )  =  ( ( ( F `  B )  x.  ( G `  B ) )  -  ( ( F `  B )  x.  ( G `  A )
) ) )
122118, 121oveq12d 6293 . . . . 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 2511 . . . 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 5857 . . . . . . . 8  |-  ( z  =  A  ->  ( G `  z )  =  ( G `  A ) )
125124oveq2d 6291 . . . . . . 7  |-  ( z  =  A  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  A
) ) )
126 fveq2 5857 . . . . . . . 8  |-  ( z  =  A  ->  ( F `  z )  =  ( F `  A ) )
127126oveq2d 6291 . . . . . . 7  |-  ( z  =  A  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  A
) ) )
128125, 127oveq12d 6293 . . . . . 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 2460 . . . . . 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 6300 . . . . . 6  |-  ( ( ( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  -  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  z
) ) )  e. 
_V
131128, 129, 130fvmpt3i 5945 . . . . 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 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
) ) ) )
133 fveq2 5857 . . . . . . . 8  |-  ( z  =  B  ->  ( G `  z )  =  ( G `  B ) )
134133oveq2d 6291 . . . . . . 7  |-  ( z  =  B  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( G `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( G `  B
) ) )
135 fveq2 5857 . . . . . . . 8  |-  ( z  =  B  ->  ( F `  z )  =  ( F `  B ) )
136135oveq2d 6291 . . . . . . 7  |-  ( z  =  B  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( F `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( F `  B
) ) )
137134, 136oveq12d 6293 . . . . . 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 5945 . . . . 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 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
) ) ) )
140123, 132, 1393eqtr4d 2511 . . 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 22119 . 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 5859 . . . . . 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 5857 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  G
) `  z )  =  ( ( RR 
_D  G ) `  x ) )
144143oveq2d 6291 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( F `  B )  -  ( F `  A )
)  x.  ( ( RR  _D  G ) `
 z ) )  =  ( ( ( F `  B )  -  ( F `  A ) )  x.  ( ( RR  _D  G ) `  x
) ) )
145 fveq2 5857 . . . . . . . . 9  |-  ( z  =  x  ->  (
( RR  _D  F
) `  z )  =  ( ( RR 
_D  F ) `  x ) )
146145oveq2d 6291 . . . . . . . 8  |-  ( z  =  x  ->  (
( ( G `  B )  -  ( G `  A )
)  x.  ( ( RR  _D  F ) `
 z ) )  =  ( ( ( G `  B )  -  ( G `  A ) )  x.  ( ( RR  _D  F ) `  x
) ) )
147144, 146oveq12d 6293 . . . . . . 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 5945 . . . . . 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 2521 . . . . 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 2462 . . . 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 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( F `  B )  -  ( F `  A ) )  e.  CC )
15273ffvelrnda 6012 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  G ) `  x )  e.  CC )
153151, 152mulcld 9605 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( F `  B
)  -  ( F `
 A ) )  x.  ( ( RR 
_D  G ) `  x ) )  e.  CC )
15493adantr 465 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( G `  B )  -  ( G `  A ) )  e.  CC )
15589ffvelrnda 6012 . . . . . 6  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( ( RR  _D  F ) `  x )  e.  CC )
156154, 155mulcld 9605 . . . . 5  |-  ( (
ph  /\  x  e.  ( A (,) B ) )  ->  ( (
( G `  B
)  -  ( G `
 A ) )  x.  ( ( RR 
_D  F ) `  x ) )  e.  CC )
157153, 156subeq0ad 9929 . . . 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 2963 . 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 369    = wceq 1374    e. wcel 1762   E.wrex 2808   _Vcvv 3106    C_ wss 3469   {cpr 4022   class class class wbr 4440    |-> cmpt 4498   dom cdm 4992   ran crn 4993   -->wf 5575   ` cfv 5579  (class class class)co 6275   CCcc 9479   RRcr 9480   0cc0 9481    x. cmul 9486   RR*cxr 9616    < clt 9617    <_ cle 9618    - cmin 9794   (,)cioo 11518   [,]cicc 11521   TopOpenctopn 14666   topGenctg 14682  ℂfldccnfld 18184   intcnt 19277   -cn->ccncf 21108    _D cdv 21995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-ico 11524  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-mulg 15854  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-fbas 18180  df-fg 18181  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-ntr 19280  df-cls 19281  df-nei 19358  df-lp 19396  df-perf 19397  df-cn 19487  df-cnp 19488  df-haus 19575  df-cmp 19646  df-tx 19791  df-hmeo 19984  df-fil 20075  df-fm 20167  df-flim 20168  df-flf 20169  df-xms 20551  df-ms 20552  df-tms 20553  df-cncf 21110  df-limc 21998  df-dv 21999
This theorem is referenced by:  mvth  22121  lhop1lem  22142
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