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Theorem ovollb 18854
Description: The outer volume is a lower bound on the sum of all interval coverings of  A. (Contributed by Mario Carneiro, 15-Jun-2014.)
Hypothesis
Ref Expression
ovollb.1  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
Assertion
Ref Expression
ovollb  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  -> 
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )

Proof of Theorem ovollb
Dummy variables  f 
y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  A  C_  U. ran  ( (,)  o.  F ) )
2 ioof 10757 . . . . . . 7  |-  (,) :
( RR*  X.  RR* ) --> ~P RR
3 simpl 443 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  F : NN --> (  <_  i^i  ( RR  X.  RR ) ) )
4 inss2 3403 . . . . . . . . 9  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR  X.  RR )
5 ressxr 8892 . . . . . . . . . 10  |-  RR  C_  RR*
6 xpss12 4808 . . . . . . . . . 10  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
75, 5, 6mp2an 653 . . . . . . . . 9  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
84, 7sstri 3201 . . . . . . . 8  |-  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* )
9 fss 5413 . . . . . . . 8  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  (  <_  i^i  ( RR  X.  RR ) )  C_  ( RR*  X.  RR* ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
103, 8, 9sylancl 643 . . . . . . 7  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  F : NN --> ( RR*  X. 
RR* ) )
11 fco 5414 . . . . . . 7  |-  ( ( (,) : ( RR*  X. 
RR* ) --> ~P RR  /\  F : NN --> ( RR*  X. 
RR* ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
122, 10, 11sylancr 644 . . . . . 6  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  -> 
( (,)  o.  F
) : NN --> ~P RR )
13 frn 5411 . . . . . 6  |-  ( ( (,)  o.  F ) : NN --> ~P RR  ->  ran  ( (,)  o.  F )  C_  ~P RR )
1412, 13syl 15 . . . . 5  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  ran  ( (,)  o.  F
)  C_  ~P RR )
15 sspwuni 4003 . . . . 5  |-  ( ran  ( (,)  o.  F
)  C_  ~P RR  <->  U.
ran  ( (,)  o.  F )  C_  RR )
1614, 15sylib 188 . . . 4  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  U. ran  ( (,)  o.  F )  C_  RR )
171, 16sstrd 3202 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  A  C_  RR )
18 eqid 2296 . . . 4  |-  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) }  =  { y  e. 
RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) }
1918ovolval 18849 . . 3  |-  ( A 
C_  RR  ->  ( vol
* `  A )  =  sup ( { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) } ,  RR* ,  `'  <  ) )
2017, 19syl 15 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  -> 
( vol * `  A )  =  sup ( { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) } ,  RR* ,  `'  <  ) )
21 ssrab2 3271 . . 3  |-  { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) } 
C_  RR*
22 ovollb.1 . . . 4  |-  S  =  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  F ) )
2318, 22elovolmr 18851 . . 3  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  sup ( ran  S ,  RR* ,  <  )  e. 
{ y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) } )
24 infmxrlb 10668 . . 3  |-  ( ( { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) }  C_  RR*  /\  sup ( ran  S ,  RR* ,  <  )  e.  {
y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) } )  ->  sup ( { y  e.  RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U. ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  ,  ( ( abs 
o.  -  )  o.  f ) ) , 
RR* ,  <  ) ) } ,  RR* ,  `'  <  )  <_  sup ( ran  S ,  RR* ,  <  ) )
2521, 23, 24sylancr 644 . 2  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  ->  sup ( { y  e. 
RR*  |  E. f  e.  ( (  <_  i^i  ( RR  X.  RR ) )  ^m  NN ) ( A  C_  U.
ran  ( (,)  o.  f )  /\  y  =  sup ( ran  seq  1 (  +  , 
( ( abs  o.  -  )  o.  f
) ) ,  RR* ,  <  ) ) } ,  RR* ,  `'  <  )  <_  sup ( ran  S ,  RR* ,  <  )
)
2620, 25eqbrtrd 4059 1  |-  ( ( F : NN --> (  <_  i^i  ( RR  X.  RR ) )  /\  A  C_ 
U. ran  ( (,)  o.  F ) )  -> 
( vol * `  A )  <_  sup ( ran  S ,  RR* ,  <  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560    i^i cin 3164    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039    X. cxp 4703   `'ccnv 4704   ran crn 4706    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   supcsup 7209   RRcr 8752   1c1 8754    + caddc 8756   RR*cxr 8882    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   (,)cioo 10672    seq cseq 11062   abscabs 11735   vol
*covol 18838
This theorem is referenced by:  ovollb2lem  18863  ovolunlem1  18872  ovoliunlem1  18877  ovoliunlem2  18878  ovolscalem1  18888  uniioovol  18950  uniioombllem3  18956  uniioombllem4  18957  uniioombllem5  18958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ioo 10676  df-ico 10678  df-fz 10799  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-ovol 18840
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