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Theorem dvfvalap 14235
Description: Value and set bounds on the derivative operator. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)
Hypotheses
Ref Expression
dvval.t  |-  T  =  ( Kt  S )
dvval.k  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
Assertion
Ref Expression
dvfvalap  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  (
( S  _D  F
)  =  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  /\  ( S  _D  F )  C_  ( ( ( int `  T ) `  A
)  X.  CC ) ) )
Distinct variable groups:    w, A, x, z    w, F, x, z    w, S, x, z    x, T
Allowed substitution hints:    T( z, w)    K( x, z, w)

Proof of Theorem dvfvalap
Dummy variables  f  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-dvap 14211 . . . 4  |-  _D  =  ( s  e.  ~P CC ,  f  e.  ( CC  ^pm  s ) 
|->  U_ x  e.  ( ( int `  (
( MetOpen `  ( abs  o. 
-  ) )t  s ) ) `  dom  f
) ( { x }  X.  ( ( z  e.  { w  e. 
dom  f  |  w #  x }  |->  ( ( ( f `  z
)  -  ( f `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) )
21a1i 9 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  _D  =  ( s  e. 
~P CC ,  f  e.  ( CC  ^pm  s )  |->  U_ x  e.  ( ( int `  (
( MetOpen `  ( abs  o. 
-  ) )t  s ) ) `  dom  f
) ( { x }  X.  ( ( z  e.  { w  e. 
dom  f  |  w #  x }  |->  ( ( ( f `  z
)  -  ( f `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) ) )
3 dvval.k . . . . . . . 8  |-  K  =  ( MetOpen `  ( abs  o. 
-  ) )
43oveq1i 5887 . . . . . . 7  |-  ( Kt  s )  =  ( (
MetOpen `  ( abs  o.  -  ) )t  s )
5 simprl 529 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  s  =  S )
65oveq2d 5893 . . . . . . . 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, 7eqtr4di 2228 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( Kt  s )  =  T )
94, 8eqtr3id 2224 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( MetOpen `  ( abs  o. 
-  ) )t  s )  =  T )
109fveq2d 5521 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( int `  ( ( MetOpen `  ( abs  o.  -  )
)t  s ) )  =  ( int `  T
) )
11 simprr 531 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  f  =  F )
1211dmeqd 4831 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  f  =  dom  F )
13 simpl2 1001 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  F : A --> CC )
1413fdmd 5374 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  F  =  A )
1512, 14eqtrd 2210 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  dom  f  =  A )
1610, 15fveq12d 5524 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( int `  (
( MetOpen `  ( abs  o. 
-  ) )t  s ) ) `  dom  f
)  =  ( ( int `  T ) `
 A ) )
1715rabeqdv 2733 . . . . . . 7  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  { w  e.  dom  f  |  w #  x }  =  {
w  e.  A  |  w #  x } )
1811fveq1d 5519 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  z )  =  ( F `  z ) )
1911fveq1d 5519 . . . . . . . . 9  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
f `  x )  =  ( F `  x ) )
2018, 19oveq12d 5895 . . . . . . . 8  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( f `  z
)  -  ( f `
 x ) )  =  ( ( F `
 z )  -  ( F `  x ) ) )
2120oveq1d 5892 . . . . . . 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
) ) )
2217, 21mpteq12dv 4087 . . . . . 6  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
z  e.  { w  e.  dom  f  |  w #  x }  |->  ( ( ( f `  z
)  -  ( f `
 x ) )  /  ( z  -  x ) ) )  =  ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) )
2322oveq1d 5892 . . . . 5  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  (
( z  e.  {
w  e.  dom  f  |  w #  x }  |->  ( ( ( f `
 z )  -  ( f `  x
) )  /  (
z  -  x ) ) ) lim CC  x
)  =  ( ( z  e.  { w  e.  A  |  w #  x }  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) )
2423xpeq2d 4652 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  ( { x }  X.  ( ( z  e. 
{ w  e.  dom  f  |  w #  x }  |->  ( ( ( f `  z )  -  ( f `  x ) )  / 
( z  -  x
) ) ) lim CC  x ) )  =  ( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) ) )
2516, 24iuneq12d 3912 . . 3  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  ( s  =  S  /\  f  =  F ) )  ->  U_ x  e.  ( ( int `  (
( MetOpen `  ( abs  o. 
-  ) )t  s ) ) `  dom  f
) ( { x }  X.  ( ( z  e.  { w  e. 
dom  f  |  w #  x }  |->  ( ( ( f `  z
)  -  ( f `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) )  =  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) ) )
26 simpr 110 . . . 4  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  s  =  S )  ->  s  =  S )
2726oveq2d 5893 . . 3  |-  ( ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  /\  s  =  S )  ->  ( CC  ^pm  s
)  =  ( CC 
^pm  S ) )
28 simp1 997 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  S  C_  CC )
29 cnex 7937 . . . . 5  |-  CC  e.  _V
3029elpw2 4159 . . . 4  |-  ( S  e.  ~P CC  <->  S  C_  CC )
3128, 30sylibr 134 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  S  e.  ~P CC )
3229a1i 9 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  CC  e.  _V )
33 simp2 998 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  F : A --> CC )
34 simp3 999 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  A  C_  S )
35 elpm2r 6668 . . . 4  |-  ( ( ( CC  e.  _V  /\  S  e.  ~P CC )  /\  ( F : A
--> CC  /\  A  C_  S ) )  ->  F  e.  ( CC  ^pm 
S ) )
3632, 31, 33, 34, 35syl22anc 1239 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  F  e.  ( CC  ^pm  S
) )
373cntoptopon 14117 . . . . . . . . 9  |-  K  e.  (TopOn `  CC )
38 resttopon 13756 . . . . . . . . 9  |-  ( ( K  e.  (TopOn `  CC )  /\  S  C_  CC )  ->  ( Kt  S )  e.  (TopOn `  S ) )
3937, 28, 38sylancr 414 . . . . . . . 8  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( Kt  S )  e.  (TopOn `  S ) )
407, 39eqeltrid 2264 . . . . . . 7  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  T  e.  (TopOn `  S )
)
41 topontop 13599 . . . . . . 7  |-  ( T  e.  (TopOn `  S
)  ->  T  e.  Top )
4240, 41syl 14 . . . . . 6  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  T  e.  Top )
43 toponuni 13600 . . . . . . . 8  |-  ( T  e.  (TopOn `  S
)  ->  S  =  U. T )
4440, 43syl 14 . . . . . . 7  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  S  =  U. T )
4534, 44sseqtrd 3195 . . . . . 6  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  A  C_ 
U. T )
46 eqid 2177 . . . . . . 7  |-  U. T  =  U. T
4746ntropn 13702 . . . . . 6  |-  ( ( T  e.  Top  /\  A  C_  U. T )  ->  ( ( int `  T ) `  A
)  e.  T )
4842, 45, 47syl2anc 411 . . . . 5  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  (
( int `  T
) `  A )  e.  T )
49 xpexg 4742 . . . . 5  |-  ( ( ( ( int `  T
) `  A )  e.  T  /\  CC  e.  _V )  ->  ( ( ( int `  T
) `  A )  X.  CC )  e.  _V )
5048, 32, 49syl2anc 411 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  (
( ( int `  T
) `  A )  X.  CC )  e.  _V )
51 limccl 14213 . . . . . . . . 9  |-  ( ( z  e.  { w  e.  A  |  w #  x }  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x )  C_  CC
52 xpss2 4739 . . . . . . . . 9  |-  ( ( ( z  e.  {
w  e.  A  |  w #  x }  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x )  C_  CC  ->  ( { x }  X.  ( ( z  e.  { w  e.  A  |  w #  x }  |->  ( ( ( F `  z )  -  ( F `  x ) )  / 
( z  -  x
) ) ) lim CC  x ) )  C_  ( { x }  X.  CC ) )
5351, 52ax-mp 5 . . . . . . . 8  |-  ( { x }  X.  (
( z  e.  {
w  e.  A  |  w #  x }  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) 
C_  ( { x }  X.  CC )
5453rgenw 2532 . . . . . . 7  |-  A. x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( {
x }  X.  CC )
55 ss2iun 3903 . . . . . . 7  |-  ( A. x  e.  ( ( int `  T ) `  A ) ( { x }  X.  (
( z  e.  {
w  e.  A  |  w #  x }  |->  ( ( ( F `  z
)  -  ( F `
 x ) )  /  ( z  -  x ) ) ) lim
CC  x ) ) 
C_  ( { x }  X.  CC )  ->  U_ x  e.  (
( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  C_  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC ) )
5654, 55ax-mp 5 . . . . . 6  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  C_  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC )
57 iunxpconst 4688 . . . . . 6  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  CC )  =  ( ( ( int `  T
) `  A )  X.  CC )
5856, 57sseqtri 3191 . . . . 5  |-  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( (
( int `  T
) `  A )  X.  CC )
5958a1i 9 . . . 4  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  C_  ( (
( int `  T
) `  A )  X.  CC ) )
6050, 59ssexd 4145 . . 3  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) )  e.  _V )
612, 25, 27, 31, 36, 60ovmpodx 6003 . 2  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( S  _D  F )  = 
U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  x }  |->  ( ( ( F `
 z )  -  ( F `  x ) )  /  ( z  -  x ) ) ) lim CC  x ) ) )
6261, 59eqsstrd 3193 . 2  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  ( S  _D  F )  C_  ( ( ( int `  T ) `  A
)  X.  CC ) )
6361, 62jca 306 1  |-  ( ( S  C_  CC  /\  F : A --> CC  /\  A  C_  S )  ->  (
( S  _D  F
)  =  U_ x  e.  ( ( int `  T
) `  A )
( { x }  X.  ( ( z  e. 
{ w  e.  A  |  w #  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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   A.wral 2455   {crab 2459   _Vcvv 2739    C_ wss 3131   ~Pcpw 3577   {csn 3594   U.cuni 3811   U_ciun 3888   class class class wbr 4005    |-> cmpt 4066    X. cxp 4626   dom cdm 4628    o. ccom 4632   -->wf 5214   ` cfv 5218  (class class class)co 5877    e. cmpo 5879    ^pm cpm 6651   CCcc 7811    - cmin 8130   # cap 8540    / cdiv 8631   abscabs 11008   ↾t crest 12693   MetOpencmopn 13530   Topctop 13582  TopOnctopon 13595   intcnt 13678   lim CC climc 14208    _D cdv 14209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-map 6652  df-pm 6653  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-xneg 9774  df-xadd 9775  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-rest 12695  df-topgen 12714  df-psmet 13532  df-xmet 13533  df-met 13534  df-bl 13535  df-mopn 13536  df-top 13583  df-topon 13596  df-bases 13628  df-ntr 13681  df-limced 14210  df-dvap 14211
This theorem is referenced by:  eldvap  14236  dvbssntrcntop  14238
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