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Theorem dvfval 21504
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 21474 . . . 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 6209 . . . . . . 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 6215 . . . . . . . 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 2513 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( Kt  s )  =  T )
94, 8syl5eqr 2509 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( TopOpen ` fld )t  s )  =  T )
109fveq2d 5802 . . . . 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 5149 . . . . . 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 5670 . . . . . . 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 2495 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  f  =  A )
1710, 16fveq12d 5804 . . . 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 3580 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( dom  f  \  { x } )  =  ( A  \  { x } ) )
1911fveq1d 5800 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  z )  =  ( F `  z ) )
2011fveq1d 5800 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  x )  =  ( F `  x ) )
2119, 20oveq12d 6217 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( f `  z
)  -  ( f `
 x ) )  =  ( ( F `
 z )  -  ( F `  x ) ) )
2221oveq1d 6214 . . . . . . 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 4477 . . . . . 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 6214 . . . . 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 4971 . . . 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 4303 . . 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 6215 . . 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 9473 . . . . 5  |-  CC  e.  _V
3130elpw2 4563 . . . 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 7339 . . . 4  |-  ( ( ( CC  e.  _V  /\  S  e.  ~P CC )  /\  ( F : A
--> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm 
S ) )
3733, 32, 34, 35, 36syl22anc 1220 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  F  e.  ( CC  ^pm  S
) )
38 limccl 21482 . . . . . . . . 9  |-  ( ( z  e.  ( A 
\  { x }
)  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) ) lim CC  x )  C_  CC
39 xpss2 5056 . . . . . . . . 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 2899 . . . . . . 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 4293 . . . . . . 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 5002 . . . . . 6  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC )  =  ( ( ( int `  T
) `  A )  X.  CC )
4543, 44sseqtri 3495 . . . . 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 5808 . . . . . 6  |-  ( ( int `  T ) `
 A )  e. 
_V
4847, 30xpex 6617 . . . . 5  |-  ( ( ( int `  T
) `  A )  X.  CC )  e.  _V
4948ssex 4543 . . . 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 6326 . 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 3497 . 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 1370    e. wcel 1758   A.wral 2798   _Vcvv 3076    \ cdif 3432    C_ wss 3435   ~Pcpw 3967   {csn 3984   U_ciun 4278    |-> cmpt 4457    X. cxp 4945   dom cdm 4947   -->wf 5521   ` cfv 5525  (class class class)co 6199    |-> cmpt2 6201    ^pm cpm 7324   CCcc 9390    - cmin 9705    / cdiv 10103   ↾t crest 14477   TopOpenctopn 14478  ℂfldccnfld 17942   intcnt 18752   lim CC climc 21469    _D cdv 21470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4510  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-oadd 7033  df-er 7210  df-map 7325  df-pm 7326  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-fi 7771  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-4 10492  df-5 10493  df-6 10494  df-7 10495  df-8 10496  df-9 10497  df-10 10498  df-n0 10690  df-z 10757  df-dec 10866  df-uz 10972  df-q 11064  df-rp 11102  df-xneg 11199  df-xadd 11200  df-xmul 11201  df-fz 11554  df-seq 11923  df-exp 11982  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-struct 14293  df-ndx 14294  df-slot 14295  df-base 14296  df-plusg 14369  df-mulr 14370  df-starv 14371  df-tset 14375  df-ple 14376  df-ds 14378  df-unif 14379  df-rest 14479  df-topn 14480  df-topgen 14500  df-psmet 17933  df-xmet 17934  df-met 17935  df-bl 17936  df-mopn 17937  df-cnfld 17943  df-top 18634  df-bases 18636  df-topon 18637  df-topsp 18638  df-cnp 18963  df-xms 20026  df-ms 20027  df-limc 21473  df-dv 21474
This theorem is referenced by:  eldv  21505  dvbssntr  21507
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