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Theorem dvfval 19737
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 19707 . . . 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 6050 . . . . . . 7  |-  ( Kt  s )  =  ( (
TopOpen ` fld )t  s )
5 simprl 733 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  s  =  S )
65oveq2d 6056 . . . . . . . 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 2454 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( Kt  s )  =  T )
94, 8syl5eqr 2450 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( TopOpen ` fld )t  s )  =  T )
109fveq2d 5691 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( int `  ( ( TopOpen ` fld )t  s
) )  =  ( int `  T ) )
11 simprr 734 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  f  =  F )
1211dmeqd 5031 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  f  =  dom  F )
13 simpl2 961 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  F : A --> CC )
14 fdm 5554 . . . . . . 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 2436 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  f  =  A )
1710, 16fveq12d 5693 . . . 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 3424 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( dom  f  \  { x } )  =  ( A  \  { x } ) )
1911fveq1d 5689 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  z )  =  ( F `  z ) )
2011fveq1d 5689 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  x )  =  ( F `  x ) )
2119, 20oveq12d 6058 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( f `  z
)  -  ( f `
 x ) )  =  ( ( F `
 z )  -  ( F `  x ) ) )
2221oveq1d 6055 . . . . . . 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 4247 . . . . . 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 6055 . . . . 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 4861 . . . 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 4077 . . 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 448 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  s  =  S )  ->  s  =  S )
2827oveq2d 6056 . . 3  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  s  =  S )  ->  ( CC  ^pm  s
)  =  ( CC 
^pm  S ) )
29 simp1 957 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  S  C_  CC )
30 cnex 9027 . . . . 5  |-  CC  e.  _V
3130elpw2 4324 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3229, 31sylibr 204 . . 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 958 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  F : A --> CC )
35 simp3 959 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  A  C_  S )
36 elpm2r 6993 . . . 4  |-  ( ( ( CC  e.  _V  /\  S  e.  ~P CC )  /\  ( F : A
--> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm 
S ) )
3733, 32, 34, 35, 36syl22anc 1185 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  F  e.  ( CC  ^pm  S
) )
38 limccl 19715 . . . . . . . . 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 8 . . . . . . . 8  |-  ( { x }  X.  (
( z  e.  ( A  \  { x } )  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) 
C_  ( { x }  X.  CC )
4140rgenw 2733 . . . . . . 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 4068 . . . . . . 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 8 . . . . . 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 4893 . . . . . 6  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC )  =  ( ( ( int `  T
) `  A )  X.  CC )
4543, 44sseqtri 3340 . . . . 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 5701 . . . . . 6  |-  ( ( int `  T ) `
 A )  e. 
_V
4847, 30xpex 4949 . . . . 5  |-  ( ( ( int `  T
) `  A )  X.  CC )  e.  _V
4948ssex 4307 . . . 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 6159 . 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 3342 . 2  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( S  _D  F )  C_  ( ( ( int `  T ) `  A
)  X.  CC ) )
5351, 52jca 519 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   ~Pcpw 3759   {csn 3774   U_ciun 4053    e. cmpt 4226    X. cxp 4835   dom cdm 4837   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042    ^pm cpm 6978   CCcc 8944    - cmin 9247    / cdiv 9633   ↾t crest 13603   TopOpenctopn 13604  ℂfldccnfld 16658   intcnt 17036   lim CC climc 19702    _D cdv 19703
This theorem is referenced by:  eldv  19738  dvbssntr  19740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-fi 7374  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-q 10531  df-rp 10569  df-xneg 10666  df-xadd 10667  df-xmul 10668  df-fz 11000  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-plusg 13497  df-mulr 13498  df-starv 13499  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-rest 13605  df-topn 13606  df-topgen 13622  df-psmet 16649  df-xmet 16650  df-met 16651  df-bl 16652  df-mopn 16653  df-cnfld 16659  df-top 16918  df-bases 16920  df-topon 16921  df-topsp 16922  df-cnp 17246  df-xms 18303  df-ms 18304  df-limc 19706  df-dv 19707
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