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Theorem dvfval 21352
Description: Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
dvval.t  |-  T  =  ( Kt  S )
dvval.k  |-  K  =  ( TopOpen ` fld )
Assertion
Ref Expression
dvfval  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  (
( S  _D  F
)  =  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  /\  ( S  _D  F )  C_  ( ( ( int `  T ) `  A
)  X.  CC ) ) )
Distinct variable groups:    x, z, A    x, F, z    x, K, z    x, S, z   
x, T
Allowed substitution hint:    T( z)

Proof of Theorem dvfval
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dv 21322 . . . 4  |-  _D  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s ) 
|->  U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) ) )
21a1i 11 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  _D  =  ( s  e. 
~P CC ,  f  e.  ( CC  ^pm  s )  |->  U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) ) ) )
3 dvval.k . . . . . . . 8  |-  K  =  ( TopOpen ` fld )
43oveq1i 6096 . . . . . . 7  |-  ( Kt  s )  =  ( (
TopOpen ` fld )t  s )
5 simprl 755 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  s  =  S )
65oveq2d 6102 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( Kt  s )  =  ( Kt  S ) )
7 dvval.t . . . . . . . 8  |-  T  =  ( Kt  S )
86, 7syl6eqr 2488 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( Kt  s )  =  T )
94, 8syl5eqr 2484 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( TopOpen ` fld )t  s )  =  T )
109fveq2d 5690 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( int `  ( ( TopOpen ` fld )t  s
) )  =  ( int `  T ) )
11 simprr 756 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  f  =  F )
1211dmeqd 5037 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  f  =  dom  F )
13 simpl2 992 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  F : A --> CC )
14 fdm 5558 . . . . . . 7  |-  ( F : A --> CC  ->  dom 
F  =  A )
1513, 14syl 16 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  F  =  A )
1612, 15eqtrd 2470 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  f  =  A )
1710, 16fveq12d 5692 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )  =  ( ( int `  T ) `  A
) )
1816difeq1d 3468 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( dom  f  \  { x } )  =  ( A  \  { x } ) )
1911fveq1d 5688 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  z )  =  ( F `  z ) )
2011fveq1d 5688 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  x )  =  ( F `  x ) )
2119, 20oveq12d 6104 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( f `  z
)  -  ( f `
 x ) )  =  ( ( F `
 z )  -  ( F `  x ) ) )
2221oveq1d 6101 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( ( f `  z )  -  (
f `  x )
)  /  ( z  -  x ) )  =  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) )
2318, 22mpteq12dv 4365 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
z  e.  ( dom  f  \  { x } )  |->  ( ( ( f `  z
)  -  ( f `
 x ) )  /  ( z  -  x ) ) )  =  ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) )
2423oveq1d 6101 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( z  e.  ( dom  f  \  {
x } )  |->  ( ( ( f `  z )  -  (
f `  x )
)  /  ( z  -  x ) ) ) lim CC  x )  =  ( ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )
2524xpeq2d 4859 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  =  ( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) ) )
2617, 25iuneq12d 4191 . . 3  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  U_ x  e.  ( ( int `  (
( TopOpen ` fld )t  s ) ) `
 dom  f )
( { x }  X.  ( ( z  e.  ( dom  f  \  { x } ) 
|->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
) )  =  U_ x  e.  ( ( int `  T ) `  A ) ( { x }  X.  (
( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) )
27 simpr 461 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  s  =  S )  ->  s  =  S )
2827oveq2d 6102 . . 3  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  s  =  S )  ->  ( CC  ^pm  s
)  =  ( CC 
^pm  S ) )
29 simp1 988 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  S  C_  CC )
30 cnex 9355 . . . . 5  |-  CC  e.  _V
3130elpw2 4451 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3229, 31sylibr 212 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  S  e.  ~P CC )
3330a1i 11 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  CC  e.  _V )
34 simp2 989 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  F : A --> CC )
35 simp3 990 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  A  C_  S )
36 elpm2r 7222 . . . 4  |-  ( ( ( CC  e.  _V  /\  S  e.  ~P CC )  /\  ( F : A
--> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm 
S ) )
3733, 32, 34, 35, 36syl22anc 1219 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  F  e.  ( CC  ^pm  S
) )
38 limccl 21330 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  { x }
)  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) ) lim CC  x )  C_  CC
39 xpss2 4944 . . . . . . . . 9  |-  ( ( ( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x )  C_  CC  ->  ( { x }  X.  ( ( z  e.  ( A  \  { x } ) 
|->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( {
x }  X.  CC ) )
4038, 39ax-mp 5 . . . . . . . 8  |-  ( { x }  X.  (
( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) 
C_  ( { x }  X.  CC )
4140rgenw 2778 . . . . . . 7  |-  A. x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( {
x }  X.  CC )
42 ss2iun 4181 . . . . . . 7  |-  ( A. x  e.  ( ( int `  T ) `  A ) ( { x }  X.  (
( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) 
C_  ( { x }  X.  CC )  ->  U_ x  e.  (
( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  C_  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC ) )
4341, 42ax-mp 5 . . . . . 6  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  C_  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC )
44 iunxpconst 4890 . . . . . 6  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC )  =  ( ( ( int `  T
) `  A )  X.  CC )
4543, 44sseqtri 3383 . . . . 5  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( (
( int `  T
) `  A )  X.  CC )
4645a1i 11 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( (
( int `  T
) `  A )  X.  CC ) )
47 fvex 5696 . . . . . 6  |-  ( ( int `  T ) `
 A )  e. 
_V
4847, 30xpex 6503 . . . . 5  |-  ( ( ( int `  T
) `  A )  X.  CC )  e.  _V
4948ssex 4431 . . . 4  |-  ( U_ x  e.  ( ( int `  T ) `  A ) ( { x }  X.  (
( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) 
C_  ( ( ( int `  T ) `
 A )  X.  CC )  ->  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  e.  _V )
5046, 49syl 16 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  e.  _V )
512, 26, 28, 32, 37, 50ovmpt2dx 6212 . 2  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( S  _D  F )  = 
U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) ) )
5251, 46eqsstrd 3385 . 2  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( S  _D  F )  C_  ( ( ( int `  T ) `  A
)  X.  CC ) )
5351, 52jca 532 1  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  (
( S  _D  F
)  =  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e.  ( A  \  {
x } )  |->  ( ( ( F `  z )  -  ( F `  x )
)  /  ( z  -  x ) ) ) lim CC  x ) )  /\  ( S  _D  F )  C_  ( ( ( int `  T ) `  A
)  X.  CC ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2710   _Vcvv 2967    \ cdif 3320    C_ wss 3323   ~Pcpw 3855   {csn 3872   U_ciun 4166    e. cmpt 4345    X. cxp 4833   dom cdm 4835   -->wf 5409   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088    ^pm cpm 7207   CCcc 9272    - cmin 9587    / cdiv 9985   ↾t crest 14351   TopOpenctopn 14352  ℂfldccnfld 17798   intcnt 18601   lim CC climc 21317    _D cdv 21318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-fi 7653  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-rp 10984  df-xneg 11081  df-xadd 11082  df-xmul 11083  df-fz 11430  df-seq 11799  df-exp 11858  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-plusg 14243  df-mulr 14244  df-starv 14245  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-rest 14353  df-topn 14354  df-topgen 14374  df-psmet 17789  df-xmet 17790  df-met 17791  df-bl 17792  df-mopn 17793  df-cnfld 17799  df-top 18483  df-bases 18485  df-topon 18486  df-topsp 18487  df-cnp 18812  df-xms 19875  df-ms 19876  df-limc 21321  df-dv 21322
This theorem is referenced by:  eldv  21353  dvbssntr  21355
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